The diameter of our galaxy is about 100,000 light years. Unless you've got some way to go faster than light then you're 2 orders of magnitude out on your estimate.
Aside from many other technical challenges (including how to shield your super-fast ship from microasteroids so it doesn't get holes punched in it at 0.5c), the main problem is: where do you get the energy needed to maintain constant 1g acceleration?
You are also forgetting how hard it to maintain that 1g of acceleration for a long period.