From Nukus to Aktau or Aqtau (Kazakhstan)

As far as I'm aware, the only sensible (possibly even only possible) way to do this trip is by train (unless you have your own vehicle). There are international trains (generally Russia-bound) heading from Nukus across the border into Kazakhstan. I'm not sure exactly what their frequency is, but I would guess at least 2-3 times a week. To get to Aktau, you'll almost definitely have to switch trains at some middle-of-nowhere rail junction in Kazakhstan. But there should be at least one train a day from here down to Aktau.

A couple of quick notes: keep a low profile in Nukus. I was spied on by OVIR while in town. Also note that the train station in Aktau is quite a long way from the city center. There's a city bus (which will be packed, and takes about an hour), or alternatively a taxi, which will seem shockingly expensive after Uzbekistan.

Giora

There are direct trains from Kungrad (100km from Nukus) to Beyneu in Kazakhstan every day, with connections to Mangyshlak, 12 km from Aktau. One of them goes all the way to Mangyshlak, probably around 3 times a week. Nukus-Kungrad is easy on a marshutka. There is no hotel in Kungrad but the police officer at the railway station called a restaurant just next to it that is also a guest house.

There is also on the schedule one train a week Nukus-Beyneu, coming from Tashkent and heading to St. Petersburg. It leaves Tashkent on Sunday night, pases Nukus monday evening, arrives in Beineu Tuesday morning and St. Petersburg on Friday morning, but this is only according to the schedule...

Shared taxi from the station in Mangyshlak to Aktau close to downtown is 300 tenge (2$) as of Dec. 2011.