Depends on what you like.

I'm not into it, but plenty of clubs open -- Marquee, Pure, LAX, etc...

Same with gambling -- pick any casino.

The following are actually options that I enjoy.

For freebie tourist activities at that time, should still be able to catch a fountain show outside Bellagio, then head inside to check out the arboretum.

If you want a bar/lounge near the airport with an awesome view of the Strip, try Mix on the 54th floor of The Hotel (at Mandalay Bay).

For dueling piano bars (not sure show will still be happening once you're there... can't remember show times), near the airport is the Times Square Bar inside New York, NY, LV (bar closes at 2:30AM). That's probably your best bet coming in after midnight, but another decent dueling piano show done by gorgeous twin sisters at Harrahs on the Strip, and Napolean's in Paris LV has a piano show too, but with better wine options than the other 2 venues.