Your questions are a little unclear. Let me take them seperately:
(1) The question in the title, "Does the angular velocity of the sun around its axis affect gravity?", has the answer "yes, slightly". In general relativity curvature of spacetime is directly proportional to the stress-energy-momentum tensor, which describes how energy is moving through spacetime. A rotating body has a slightly different energy and momentum configuration than a non-rotating one, and this causes an effect known as "frame dragging". But it's a very tiny effect, for all practical purposes we can ignore the rotation of the sun when calculating its gravitational effects.
(2) In the body of your question you write "Spacetime is flat for me but I can calculate that the sun should bend/accelerate towards me (or I towards the sun)." That's just wrong. The curvature of spacetime is a physical thing, and you cannot make it disappear by changing coordinates (i.e. reference frames). So no, spacetime isn't flat for you. The curvature of spacetime is the same whether you regard yourself as moving or stationary. The coordinates you give to things may be slightly different in the two cases, but the net result (spacetime is curved by the sun) is the same either way.
(3) You say "... from my perspective, the sun is spinning so fast around its own axis, that acceleration is not happening." Why in the world do you think that? Are you trying to use some kind of coordinate system in which the Earth is stationary and non-rotating? If so those are non-inertial coordinates, and trying to do physics in a non-inertial coordinate system is very hard and can lead you to all sorts of mistaken notions. If you do the math carefully you'll get the same result for the attraction of gravity in those coordinates, but it's much harder to do the math (and trying to use intuitive shortcuts will trip you up).
(4) Your final question is "Would it be possible for me to spin really fast around my own axis to ignore the sun's gravity?" and the answer is "no, absolutely not". The sun's gravity is a real, physical thing and you can't "ignore" it. If you spun yourself fast enough you could perhaps slightly increase the attraction between yourself and the sun (because you'd have more energy) but you'd actually have to spin so fast that'd you'd vaporize/fly apart long before the effect was measurable.
