It's not completely wireless; to get any reasonable bandwidth out to the users, you need fiber to the towers, not just T1 or radio uplinks, but that's not too hard to do. (As another poster says, the telco's run by the government, so they shouldn't have a problem getting permits, just the usual issues with new construction in old cities.)
No reason to use old phones - the newer standards are much more efficient at spectrum usage.
And there's been fiber to the island for a long time; the problem has been that the US embargoes on trade with Cuba severely limited the services the telcos could provide. To the extent that that was caused by Treasury regulations (which Obama can change for two years) rather than law (which requires the Republicans in Congress to cooperate), they can get some of that service running quickly.