If by “ancient Latin alphabet” you mean the alphabet as depicted in charts like this you’re talking about the Archaic alphabet, not the alphabet the Romans used for Classical Latin. The Romans after the Archaic Period used the same alphabet as we do (with minor additions depending on our precise European language), at least in inscriptions–Roman cursive is very different in form. The charts you’re looking at are very misleading, in that Latin was written in the Archaic Period either right to left or boustrephedon, alternating direction with each line. But these are only the very earliest Latin inscriptions. By the time Latin really starts to be used regularly as a written language it is being written left to right, with the letters oriented to suit.