DWM (WWI) and Mauser (WWII) made the best wartime Lugers. As Sniper said a good one with matching numbers will be above your suggested budget. You could get a mixed parts "shooter" in your price range. I have a pristine 1941 byf (Mauser) that I've never shot and it probably doesn't have much over 100 rounds the tube ever. I also have a 1913 dated DWM with parts from at least 15 other Lugers of the era - and it shoots great with Federal Champion. I've had it since 1975 and after thousands of rounds only one part broke, which fortunately was an easy fix. Guess which Luger is more fun!
Of the post war Lugers, the best is the new Mauser that was produced in the late 1960s until maybe 1980+. Some aren't true German Army P08 copies but some are. Also you'll sometimes see the Mitchell Arms stainless Luger lookalike but I understand galling and other problems afflict them. Whatever you do, don't buy a Stoeger Luger .22. I had one that was a jamomatic and so did everyone else. The Erma .22s looked more like a real Luger and worked better than the Stoeger.
Check out the Luger forums for people who really are in to these fine guns. And for good deals on a shooter.
The biggest issue with assembling an AR isn't so much getting the parts together right - it's getting the right parts together. You'll remember the quality of a gun long after you forget how much you paid for it.
US Army 1966-69, VFW Life Member, Retired Geek