if you had to walk 13 miles now to get food or water, you could and would do it.
average walking pace is approx 3mph (20mins/mile) so you could do the race in just over 4 hours with no training, assuming you don't have some underling medical condition. The world record is 58:23, which is a pace of 4:23min/mile.
So the answer to your question is yes, 3 weeks to train for a half is enough, and my guess is that you'll be somewhere between these extremes.
As for tips for training, just get out and try to run frequently, up to 5 times per week, resisting the urge to race all of these runs. in fact, make them all a comfortable, conversational pace. Do a 6mile run this weekend, but KEEP IT SLOW. You should finish feeling fresh, not knackered. Then do 8miles the next week, same easy pace, then maybe 10 miles the following week, same easy pace.
Your legs will possibly feel a little heavy after the longer runs, and might feel a bit fatigued during the next 3 weeks if you do this, but you shouldn't be exhausted. If you are, you're doing too much, or more likely, running too fast. Slow down, and if you think you're going too slow, slow down a bit more. the mileage is more important than the speed, as you say the time doesn't bother you, but you want to complete it.
And remember on race day not to head off too fast, otherwise you'll blow up a few miles down the road, and the rest of the race will be miserable. Have fun, and if you like the experience, enter another one, but make it a few months down the road and leave yourself some time to prepare for it, then notice the difference it makes on race day.