There’s absolutely no reason why you can’t run your first half marathon by the fall. I posted an excellent 12 week “beginners” training program on this site (2hr – Half Marathon) that will not only get you there but you’ll develop all of the confidence you need to successfully complete the run in 2 to 2 ½ hours without any strain or day after effects. One note, all of the runs in this program should be done at a pace that is 1 ½ - 2 min. slower than you would run a 5-10 k unless noted. That will help you avoid injury and allow for the development of stamina. The short runs will help you develop the speed.
Take it slow and listen to your body, if it tells you to slow down, slow down. If it tells you to stop, stop, this is only training and you don’t want to hurt yourself and not be able to train.
This program really works, I started using it a year ago and until then I had never even run a mile. I the past year I’ve completed 3 half marathons culminating with the Las Vegas run this past Dec and I’ve registered for 2 more so far for this season.
Not to bad for a guy in his 60’s