I'm going to try to explain my process in answering, to give you some resources for coming up with more names in the future.
(Also, note that I like to mark my long vowels, "ā ē ī ō ū". Many people don't do this; it won't be any less correct if you remove them when writing your names. They represent a pronunciation difference which disappeared in later Latin.)
Lightspeed
One of my favorite resources for English-to-Latin is Ludwig von Döderlein's Hand-book of Latin Synonymes [sic]. It lists the subtle meaning differences between synonyms, as well as acting as a limited thesaurus.
Under citus (fast):
CITUS; CELER; VELOX; PERNIX; PROPERUS; FESTINUS.
1. Citus and celer denote swiftness, merely as quick motion, in opp. to tardus, Cic. Or. iii. 57. Sall. Cat. 15. Cic. Fin. v. 11. N. D. ii. 20. Rosc. Com. 11. Top. 44; velox and pernix, nimbleness, as bodily strength and activity, in opp. to lentus; properus and festinus, haste, as the will to reach a certain point in the shortest time, in opp. to segnis Gell. x. 11. 2. Citus denotes a swift and lively motion, approaching to vegetus; celer, an eager and impetuous motion, approaching to rapidus. 3. Pernicitas is, in general, dexterity and activity in all bodily movements, in hopping, climbing, and vaulting; but velocitas, especially in running, flying, and swimming, and so forth. Plaut. Mil. iii. 1, 36. Clare oculis video, pernix sum manibus, pedibus mobilis. Virg. Æn. iv. 180. Curt. vii. 7, 53. Equorum velocitati par est hominum pernicitas. 4. Properus, properare, denote the haste which, from energy, sets out rapidly to reach a certain point, in opp. to cessare; whereas festinus, festinare, denote the haste which springs from impatience, and borders upon precipitation. (ii. 144.)
From personal experience, I've also heard vēlōx described as "moving quickly right now" and celer as "able to move quickly", so Usain Bolt sitting on the couch would be celer but not vēlōx. But Döderlein doesn't mention this, and actually implies the opposite; I'll ask a new question about that.
Then under lūmen (light):
LUMEN; LUX.
Lumen (λευσσόμενον) is a luminous body, like φέγγος; lux (λευκή) a streaming mass of light, like φάος. Cic. Fin. iii. 14, 45. Ut obscuratur et offunditur luce solis lumen lucernæ. Curt. viii. 2, 21. Sed aditus specus accipit lucem; interiora nisi allato lumine obscura sunt. Cic. Acad. iv. 8, 28. Si ista vera sunt, ratio omnis tollitur quasi quædam _lux lumen_que vitæ; that is, reason alone is in itself bright and light, and at the same time spreads brightness and light over life. Also, in a figurative sense, lumen denotes distinction, lux only clearness. Cicero (Man. 5.) calls Corinth, Græciæ totius lumen, but Rome (Catil. iv. 6.) Lucem orbis terrarum; Corinth is compared to a glimmering point of light; Rome is distinguished as that city in comparison with which all other cities lie in darkness. (ii. 66.)
So I'd say celer (moving very quickly) and lūx (light itself, as opposed to a source of light) are the words you'd want.
Combining them unfortunately requires significant knowledge of Latin grammar; there's no easy way to do this step without being familiar with the language. In this case, I would say Celeritās Lūcis, literally "the swiftness of the light".
Joonas also suggested that vēlōx might fit the meaning of "lightspeed" better: light really can't move slower than its maximum speed, and the 'running' meaning sounds more like what a "lightspeed" character would have. In this case, it would be Vēlōcitās Lūcis, "the velocity of the light". This is also more obviously related to speed, for non-Latin-speakers.
Bull's Eye
This is an idiom in English. Oculus bovis would be a literal translation, but there doesn't seem to be much precedent for using oculus like this. When looking for how a specific word was used, I usually put it into the Perseus Word Study Tool, which then brings up the entry from Lewis and Short's Latin dictionary. (I could also search in L&S directly, but the Word Study Tool has a nicer interface.)
In this case, L&S lists plenty of idioms involving eyes, but none seem quite right. Most focus on seeing or perceiving, rather than precision or accuracy.
The closest idiom I can think of is acū tetigistī, literally "you touched it with a needle" (used to mean "you're exactly right"). Changing the grammatical form, Acū Tangō is "I touch it with a needle", or idiomatically "I hit it precisely".
Holy Judgement
For this one I'm looking at Christian sources, since for me "holy" is more strongly associated with Christian religion (I'd use "sacred" instead if referring to classical Roman religion).
The Diēs Īrae in particular describes the Day of Judgement repeatedly, always using the words jūdex, jūdicō, jūdicāns...
So I would go with Jūdex Sacer (the holy judge) or Jūdicāns Sacer (the holy judging person). Or more literally, Jūdicium Sacrum (the holy judgement).
Godlike Powers
"Godlike" is difficult. I'm translating it here as "the powers seem to come from a deity (through this mortal person)"; an alternate meaning would be "the person using these powers actually seems to be a deity". Let me know if this was your intent.
From Döderlein again:
POTENTIA; POTENTATUS; POTESTAS; VIS; ROBUR.
Potentia, potentatus, and potestas (πότνιος) denote an exterior power, which acts by means of men, and upon men; whereas vis and robur denote an interior power and strength, independent of the co-operation and good-will of others. Potentia denotes a merely factitious power, which can be exerted at will, like δύναμις; potentatus, the exterior rank of the ruler, which is acknowledged by those who are subject to him, like δυναστεία; potestas, a just and lawful power, with which a person is entrusted, like ἐξουσία. Tac. Ann. xiii. 19. Nihil tam fluxum est quam fama potentiæ non sua vi nixæ. Vis (ἴς) is the strength which shows itself in moving and attacking, as an ability to constrain others, like κράτος; robur (from ἐῤῥῶσθαι) the strength which shows itself in remaining quiet, as an ability to resist attack, and remain firm, like ῥώμη. (v. 83.)
So vīs seems like what you want. This is "power" in the sense of strength or force. "Divine" is easier, since the English word was borrowed from Latin: dīvīnus. Combined would give Vīs Dīvīna.
Armageddon
Google Translate was fairly accurate on this one. English Armageddon comes from Greek Harmagedōn from Hebrew Har Məgiddô "Mount Megiddo". So if you wanted to refer to the actual mountain, Harmagedōn would be a decent translation.
I assume you want a synonym for "apocalypse" rather than the name of a mountain in Israel. Looking up "apocalypse" isn't likely to help either, though: it comes from Greek apocalypsis, "uncovering", so the direct Latin translation would be revēlātiō. The modern meaning of "end of the world" is because the end of the world was predicted in the Book of Revelation, but it didn't have that meaning in Classical times.
Unfortunately I don't know a good Classical term for "the end of the world". To my knowledge there was no Ragnarök equivalent in Roman mythology, so the closest you could get would be a literal but artificial Exitium Mundī "the final destruction of the world". Alternatively, you could add in a religious allusion and go with Diēs Īrae "the day of wrath" (from Zepheniah 1:15), which is a well-known term in Christianity but also has a good literal meaning. Or, if you aren't concerned about the ancient meaning, take the loanword Apocalypsis. (Or for additional foreign-ness, use the Greek spelling: Ἀποκάλυψις.)
