I don't know about the First Century, but we do not have a tradition to bury a man in his tefillin, but actually wrapped up in his tallit. In Ashkenazi circles, we also bury a man in his kittel. Although I do not have a source, I'm fairly certain it was not the custom either in the First Century to bury a man in his tefillin.
As for Jesus donning tefillin while being crucified (or donning tefillin period), I am far from a expert on Christianity. But it seems to me that most people who were crucified by the Romans were executed naked (despite the modest church renditions of the crucifixion), and the Romans would not have allowed someone to wear something like tefillin during an execution.
Another thing is that according to Jewish Law, one must maintain a clean body while wearing the tefillin. A crucified person, being both naked and bloody, would not be permitted to wear tefillin, and it would be considered both a violation and disrespect to the tefillin, a very important mitzvah.