For the trip you're doing, if you have a visa credit card with a PIN, you can use it just about everywhere to withdraw from the many cashpoint machines.

The only place you'll need cash is Uzbekistan (and plan enough for your whole stay there: accomm, food, fuel, entry fees to sights etc)
and maybe entering Mongolia, though I seem to recall we found an ATM quite quickly.
Even in a village in the middle of nowhere, as they are in Mongolia, we found a bank where we could withdraw inside the bank, not at a machine, when we had run out of local currency and they couldn't exchange dollars for some reason, or we didn't have any left: I can't quite remember!

In addition to that, it's useful to have the dollars for when you cross any small borders and there are no machines until the next major town and you may need to buy fuel or whatever beforehand.

I would go along with mark's range, you might feel more comfortable with more because of Uzbekistan.
My personal minimum is probably 500$ as the other mark said.

I also remember being able to withdraw dollars from some ATMs in Gerogia, which was very helpful.

Check caravanistan to calculate how much you'll need for visas, plus Uzbekistan, plus your personal emergency stash.
Hopefully you won't need to pay any bribes...we were lucky on that route and only had one case at the tajik border, based on our lack of prep and thinking the disinfection fee he wanted was a real one.

Have fun!