Just got back from ASB yesterday. If you're still there, the oguz kent (sofitel) has free wifi that you can use
Other thoughts:
Food at the British Pub is a miss
The restaurant at the far back of First Park isn't too bad, live music some nights too
I had no issues with the 11pm curfew, walked back from sofitel through downtown at 12am, to the hotel by 1st park. Even made a stop at the post office and said hello to many officers.

The Soviet welcome sign for Turkmenbashi is by far the best I saw, regret not stopping and snapping photos. There's a big golden statue on the highway to the mosque Turkmenbashi built outside ASB in his old village with out any police around it to take photos. The Mosque is worth checking out.