The only evidence for the existence of T. Apollonius C. f. is a single inscription. The inscription is number 2203 in the Corpus Inscriptionum Latinarum, volume I, part 2 (CIL I:2 2203). It reads:
The inscription therefore records the fact that Titus Apolonius (the spelling is a variant of "Apollonius"), son of Gaius, and Publius Babrinius, son of Marcus, were Duoviri (i.e. the highest magistrates of a Italian city). In this case the city was Aquileia: the inscription is now built into the fabric of the church of St Stephen in Aquileia.
Johannes Baptista Brusin's collection Inscriptiones Aquileiae, vol. I p. 19, also includes the inscription (number 33). Brusin comments, "Duóvirí titulum attribuunt aetátí bellum sociórum praecedentí nam postea Aquileia múnicipium factum quattuorvirós exhibet." ("The title duoviri attributes [this] to the period preceding the social war, for afterwards Aquileia, having been made a municipium, exhibits quattuorviri [in stead].")
Aquileia was made a Latin colony in 181 B.C. (Livy 39.22.6; 39.45.6; 39.54.1 - 39.55.6; 40.26.2; 40.34.2) and was reinforced with the settlement of 1,500 Roman families there in 169 (Livy 43.1.5; 43.17.1). The inscription is highly unlikely to pre-date 181 B.C., both because of the spelling of the inscription and because Aquileia would probably not have had duoviri before being a colony. So the inscription should be dated between 181 and c.90 B.C.
At some point before the lex Acilia (which is usually dated to 122 B.C.), and at any rate before the social war, magistrates of Latin colonies acquired Roman citizenship automatically by virtue of their magistracy (see Sherwin-White, The Roman Citizenship, 2nd ed., pp. 111-2). If the T. Apollonius inscription dates from between 122 and c.90 then Apollonius would therefore have become a Roman citizen when he became duovir, if he was not one already. If the inscription is older then he may not have been a citizen at the time of the inscription. Even in this case he would, as a Latin, have held the right to acquire Roman citizenship by going to live at Rome, and it is possible that he did so. Moreover, if his posterity survived beyond the social war, then they will have been awarded Roman citizenship as part of the settlement that ended the war, as were all Latins.