More like ten miles south overland (not very feasible), maybe 12 by road. Also, Christ wasn’t born in the caravansary: there was no room to find a private place there. Thus the choice to find a manger in a cave (traditionally the cave where David hid from Saul) or inside a house (typically down a level from the main floor, so that sweepings could be swept there.)
Some Biblical evidence plus tradition suggests the Last Supper was held at the inn run by John Mark’s mother. Inside Jerusalem itself (albeit near one of the gates), it wouldn’t have had room for a proper caravanserai. But that might have been its name anyway passed down from old times; or since Jesus seems to be sending His disciples out to take up arrangements He has already made, it might be a code word to let people know they’ve met the right people. Even up to the point where Iscariot left, Jesus wasn’t being clear about whether this was a Passover meal or not, and I think there are strong arguments (reconciling disparate data between the Synoptics and GosJohn) that He held the Last Supper as a Passover one night early (the lamb cooking having already started at sundown for the huge crowds involved; there’d be a big party that night at the Temple for the cooking, which would delay getting together enough Sandhedrin members for a preliminary trial, which the actual Passover meal and the Sabbath would have further hampered.) Iscariot evidently understood this to be a Passover meal – as did various Gospel authors later – and if it was one night early, that would be strong contextual evidence (though wrongly inferred) that Jesus going to start a military revolution against the regime as the Maccabees had done: rabbis or family heads only hold the Passover one night early if they already expect a life-threatening emergency the next night, like a battle!
This also explains why the Sanhedrin, despite having already resolved to put off doing something about Jesus again until after the Passover, changed their minds in such a panic after Iscariot went to them. Meanwhile, the apostles (and any other disciples around, probably including John Mark) aren’t sure whether Iscariot is going out to get the proper food for a Passover that hasn’t happened yet, or to give Passover charity in the street as the disciple in the honored/son position.