The movie tells us the last sighting of Batman was the night Dent died. There was no reported sightings of him since. Alfred said he hung up his cape and his cowl but he didn't move on, meaning he didn't move on to build a new life for himself. Not that he was still doing Batman related activities. He just stayed in a state of misery for years shut off from everyone.
The last "confirmed" sighting. I think there's a hint of ambiguity in that line, although I don't personally believe he was out in full costume between the two movies.
I do think he used the cave at some point though, probably keeping an eye on things in Gotham. Alfred's line, "You haven't been down here in a long time" confirms that Bruce had been down there doing stuff at some point or another, since that cave did not exist in the TDK-era.
Also, we have no idea when the Dent act was passed. I'd like to think the order went something like:
-Batman takes the fall for Harvey
-The new batcave is built in the new Wayne Manor, Bruce keeps an eye on things but activity is slow in the time period following TDK.
-Dent Act is passed, within a year of its institution the streets start getting cleaner
-Bruce realizes there's truly no longer a need for a Batman, and the lie to keep Dent Gotham's hero has paid off.
-Still having an itch to help where he can, he turns to philanthropy and takes on Miranda Tate's clean energy project at Wayne Enterprises.
-Pavel writes a paper about how it can be turned into a nuke, Bruce halts the project (I can just see him reading that paper in the Bat-cave)
-Bruce becomes the bitter recluse we see in TDKR and starts wasting away