I'd think that for a century you'd want to get at least Shimano 105 or the equivalent. You might consider upping the bike budget a bit. How short is short, and what's your inseam? Specialized makes some good bikes if you're on the shorter side. A Ruby, while perhaps pushing the boundaries of your budget a bit, might be good for those neck issues.

You can cheap out on jerseys, but things that you should be willing to spend on: Shorts (at least 2 pairs, so that you don't have to do laundry before every ride), gloves, saddle, shoes. The sky's the limit on what you could spend, but the question is what can you afford?