How do we know how fast surface of neutron star is rotating?
I suppose we don't know the answer with total confidence, but I'd say we think we know with good confidence. Because we have robust evidence along with a model that matches that evidence.
I understand that we can count pulses in time, but how do we know that magnetosphere is rotating along with surface of neutron star?
I suppose we don't actually know for sure. But I can't think of any process whereby a stationary spherical body can emit regular pulses of radiation.
Can we measure rotation of a neutron star in any other way like red-blue shifts? It should be pretty difficult due to small radius and big distance but is it possible? Is it already done?
No, and no. At least as far as I know. But most of what I know came from Frozen Star by George Greenstein. The first half of the book is all about neutron stars. It dates from 1984 and it could be a little out of date. But as far as I know there's not much wrong with it, and it's a good read regardless. Note that you can buy a second hand copy fairly cheaply.