The words incipientem interfici are the present active participle with the passive infinitive, and they correspond to the words in Greek μέλλοντα ἀναιρεῖσθαι, which are also a combination of a participle and a passive infinitive..
According to Thayer, among the meanings of the Greek word μέλλω is "to be on the point of doing or suffering something", so it is often translated as I am about to, so μέλλοντα ἀναιρεῖσθαι means who is about to be killed.
The basic meaning of incipio is I begin, so literally incipientem interfici means who is beginning to be killed, but since the tense of a participle is relative to the main verb, it actually means who was beginning to be killed, but it can be understood in the sense of:
who was about to be killed
However, this is not the original meaning of incipio. According to a study by Giovanbattista Galdi, Jerome's use of incipio to translate μέλλω resulted in a change of meaning in the way it was sometimes used afterwards. He said:
In our opinion, we may be confronted here by a 'loan-shift', i.e. the
process by which 'the semantic field of a lexeme in the recipient
language is adjusted to replicate that of an already partially
equivalent one in the source language' (Coleman 1975: 106).
To whatever extent such a change may have occurred, the main point is that originally incipio was a less than perfect fit for μέλλω, so concerning the translation of the New Testament, it should be understood in the sense that Jerome was trying to capture, i.e., in the original sense of the μέλλω as Thayer described it:
to be on the point of doing or suffering something
Concerning the accusative:
The participle incipientem is in the accusative along with some other words which are also in the accusative, and they all refer to Paul:
virum hunc comprehensum a Iudaeis et incipientem interfici ab eis… eripui
The are in the accusative because they serve as the direct obect of the verbe eripui, I rescued:
I rescued… this man who was seized by the Jews and who was about
to be killed by them