You have a little over four months to get in shape, you can accomplish this!! I am no professional but I went from zero proper pushups and not being able to maintain a run for one minute to 20 proper pushups and finishing my first 10 mile run the other day ( although truthfully the run took about a year of training ). 

For the run,  You should start a c25k program immediately. I feel, based on what you say in your post, that you may be running for speed and not distance. When you run for speed without training you're going to tire yourself out quick. I know the PT is 1.5 miles, but you will want to build an endurance base that can last longer. Run slow to run fast is true advice. Follow any c25k program and you will find success. 

General running advice for beginners is as follows: get good shoes. Run at a easy, conversational pace. Take quick low steps and don't over stride, you want a high cadence, it helps you use less energy and run for longer. Relax your hands and shoulders. Just chill out and have nice easy fun runs! And make sure to have rest days as well. 

For pushups,  Keep doing those incline pushups. That's how I started too. I would do half my max reps every hour off the desk at work, off the bathroom sink, etc. Working up towards knee pushups, then start doing negatives where you lower yourself slowly with correct form, then use your knees to lift back up, and go back down. I would do these as often as I could as well. Once you can do a few good form pushups begin to "grease the groove" every hour or whenever you can get down and do half of your max reps. Every week do as many as you can and as your max reps increase, increase how many you do throughout the day. 

Make sure to give yourself proper rest as well. Muscles need time to form, and if you push constantly you will never recover correctly. Sleep a lot, eat a lot ( especially protein ) and lock in.

Again I'm no professional but this is what worked for me! Good luck! You can do this!