And consider they are making their first encounter with Khan aboard a drifting sleeper ship with suitably warm breathable atmosphere, lighting and artificial gravity, (!!!) all of which works after (at least) 200 years.
I choose to believe that the cylindrical portion of the ship contained a 2001
-style centrifuge which produced the gravity.
Another thing to consider: Kirk said at the end that their heading took them near Ceti Alpha, so presumably they were somewhere in that general vicinity already. Now, Alpha Ceti (as it's more properly called) is about 250 light-years away, give or take -- and the Botany Bay
was a sublight ship. So assuming they were within, say, 20 ly of Alpha Ceti, then "Space Seed" couldn't have taken place any earlier than 2226 or so.
I assume you're talking about the main hull rather than the strap on pods. In that case the ship is nowhere big enough to produce a 1 g field. If you're talking about the pods then the whole ship would have to rotate since the pods don't form a complete circle.
Also, Ceti Alpha =/= Alpha Ceti. Trek star names don't necessarily match up with ours. I would assume that they have their own star catalog, perhaps UFC (United Federation Catalog). How many different names is Alpha Centauri known by even now?