"Any" given Roman soldier would have had next to no knowledge of Aramaic or Syriac. An Italian farm boy recruited into the legions during the late first century CE and then stationed in Spain would have virtually no contact with either language, unless he happened to live in the Syrian ex-pat neighborhood in Rome or happened to have a slave-nurse who spoke Aramaic to him when he was a kid (which actually isn't all that unlikely in the late 1st century BCE). There were certainly plenty of soldiers levied from Aramaic- or Syraic-speaking regions in this time, but the total numbers were very small compared to the total population of Roman soldiers.

In the early Empire (the "time of Jesus"), the legions stationed in the Syrian province were not, by and large, levied from Syria. Most of them were Italians, like the famous Legio X Fretensis. This one was raised by Octavian early in the civil war and fought at Naulochos, Actium, and was then at Caesarea Maritima in Judea. It was then in Syria. Judaea itself was what is sometimes called a "third-class" province, in that it was small and quiet enough to be ruled by an Imperial Prefect of equestrian rank (rather than a senator). The province had a little force commanded by the prefect, some of which was made of auxiliaries (Samarians, for instance). These guys probably spoke some Aramaic. The rest of the force were Italians or other Western Mediterranean in origin, however: Cohors Prima Italica Civium Romanorum, the Cohors Secunda Italica Civium Romanorum, and the Cohors Prima Augusta.

The "lingua Franca" of the legions was still Latin at this point, even in the east, but many of them probably had basic proficiency in koine Greek, especially those from Campania/Bay of Naples.