Short answer is "no". Orbital mechanics don't work like that. (Big disclaimer- I'm not an expert, and while what I'm about to type should be basically correct, I can't guarantee (at this time in the morning) that I haven't made some mistakes).
Your basic mistake is assuming that the bullet you're firing is stationary before you fire it, so all you're having to do is propel it towards it's target. It isn't. The bullet (and the gun, and the marksman) are all orbiting the sun at 29.8 km/s (which is the speed that the Earth is orbiting at). By "orbiting at 29.8 km/s", what we mean is "travelling in such a way as to miss the Sun at 29.8 km/s". So if you want your bullet to hit the Sun, you need to cause it to stop doing that- you need it to lose 29.8 km/s of orbital speed. I know you were only using it as a metaphor, but for reference- a bullet from a typical gun travels (i.e. changes velocity) at less than 0.5 km/s.
Counter intuitively, travelling to Alpha Centauri would be much easier (although of course it would take a long time!). Solar escape velocity starting from Earth is only 42.1 km/s, which means that you only need to at ~13 km/s before you're away from the Sun's gravitational grip. There are two reasons for this. Firstly, Earth's orbital velocity is already quite fast, so getting to escape velocity means adding a relatively small amount (albeit to get to an overall high speed). Secondly, gravity is inversely proportional to the square of the distance- that is, moving 100 km closer to the sun will increase the gravity you experience by more than moving 100 km away from the sun will decrease it. Without getting into the messy details of it, this means that the necessary changes in velocity get sharper the closer you get to the sun- hence why Earth (which seems quite close to the Sun, in the grand scheme of things) is in a stable orbit at 29.8 km/s, but could escape completely for a mere 13 km/s more.
Clear as crystal?