The cheap hotel I know of in Aktau is at:
Microrayon 3, Dom 20
I can't remember what it's called, and I think it's signed only in Kazakh ("Konak Uyi" being "Hotel" in Kazakh). Staff are very friendly, and delighted to have foreigners staying at their hotel. Room prices start at around 12 USD.
About the address: any taxi driver in Aktau should be able to find it. Aktau is divided into districts (microrayon) and all the buildings (dom) have big numbers painted on them: it's a unique address system. If you want to walk, head along the main road from the ship statue (in the direction of Microrayon 3) and turn left about 10-15 minutes later at a bank. The hotel is 100 yards down on the left. Alternatively, just find Microrayon 3 and wander around looking at building numbers until you see 20. If, for whatever reason, that hotel has closed (I was there about a year and a half ago), you should find other cheap hotels in Microrayon 3. The LP only lists hotels with prices of 60 USD and up, but cheap hotels do exist in Aktau.
Giora