Old Latin /r/ probably had non-trill allophones: at minimum, a tap/flap [ɾ], and likely approximant and fricative allophones as well.
The existence of [ɾ] in Old Latin is supported by rare cases of /d/ changing into /r/, such as meridies from medius + dies (usually explained as involving dissimilation of the two /d/ sounds) and old spellings of certain ad-prefixed words such as arfuise for adfuisse and arvorsum for adversum. The voiced stop /d/ is more acoustically similar to [ɾ] than it is to [r], so when these changes of /d/ to /r/ occurred, it's likely that the tap/flap [ɾ] was a fairly common realization of /r/, at least in these positions (between vowels and at the end of syllables).
Actuually, in modern languages that have a phoneme transcribed and described as the alveolar trill /r/, the /r/ phoneme frequently has a range of allophones that includes non-trilled sounds. This is because it requires a fair amount of articulatory "effort" to produce an acoustic alveolar trill, so any "weakening" (lenition) of the sound is liable to turn it into something other than a trill. Therefore, evidence for the existence of non-trilled pronunciations of /r/ in Old Latin doesn't rule out the presence of a fortis trill [r] as well. However, I don't know of any positive evidence for a trill in Old Latin. If a trill was used, it may have only been a minority variant, or only predominant over other allophones in specific contexts. The contexts that we would most expect to favor a trill pronunciation are at the absolute start of a word, and when /r/ was doubled (geminated) as /rr/.
In Classical Latin as well, it's plausible a trill may have been a preferred realization only in certain contexts. Many of the Romance languages have a tap/flap as one reflex of Latin /r/.
"Rhotic Variation in Tuscan Italian", by C. Celata, A. Vietti, and L. Spreafico, describes the variety of realizations of /r/ in present-day Italian, where approximant, trill, tap, and fricative realizations can all be found.

Obviously, this table has no direct applicability to Old Latin, but my impression is that Old Latin data is consistent with there being a range of realizations similar to the range we see in present-day Italian. Aside from being related languages, the phonemic and phonotactic behavior of /r/ in Old Latin is similar to that of /r/ in present-day Italian: both languages have one rhotic phoneme, /r/, which is always a singleton consonant when in word-initial position or before or after another consonant, but can be singleton (short) or geminate (long) in intervocalic position.