Originally Answered: Half-Marathon in 4 weeks?
My first reaction would be - maybe - that you could finish a HM, pretty hard to think that it could be around 1:45, since by your own admission, your longest runs have been 6 milers. Not too what your longest runs were in high school but an HM is a completly different animal.
But - since you are already apparently in good decent shape, you might be able to train your body to last 13.1 miles. There are programs online like Hal Hidgon's free training programs:
http://www.halhigdon.com/training/51131/... - you might be able to take the last 4-5 weeks of such programs and get in some decent long runs, and to do that without cramming a lot of extra mileage in a relatively short amount of time.
Why do you need/want to finish in 1:45, especially for a first-time half, and really especially with relatively little distance training? Perhaps you could either look at the race as a starter HM, to get the feel of it, and then enter a second HM later on, and run THAT race with a specific time in mind.
Disclosure - I've finished three HMs, I am admittedly very slow (my average time was 2:15), and my last and worst HM was the one that I did not put in the long runs, which IMHO are very important to train your body to stay in motion for close to 2 hours.
In any case, I wish you luck, and remember to listen to your body.