I've answered a similar question that covers some of the same points: My pushups don't increase
Why you're not progressing
If you're wondering why you're not progressing here, it's because you're doing the same thing over and over. Your body simply adapts to doing just that, and doesn't get challenged. Imagine trying to learn math by solving 5+7 over and over. At some point, we're just repeating a routine rather than actually working to figure things out. The challenge is gone, and with it goes our chance to progress to the next level.
There's a lot more than just the actual working out that goes into improving your performance. The big trifecta we usually preach is
You can't focus on two out of three here, and expect results. It needs to be all three.
Since there are some details missing with regards to the quality of your diet and sleep, I will forego that for now, and focus on the training. But keep in mind that none of these tips will serve you unless you eat well, and sleep well. If at any point you feel more fatigued than you've used to, a reduced quality of eating and sleeping habits are the usual suspects.
Neglecting the bigger picture
A common mistake is to look in the mirror, find a point you want to work on, and focusing exclusively on that. You mention you want to get some bulk on your arms by November, and I'd say that's doable. But if you only work out your upper body, this has a tendency to cause imbalances in your body, which leads to back pain, neck pain, hip pain, and from there it just branches out.
Since you have a short-term goal in mind (some arm bulk with a 4 month timeframe), I'd say it's fine to concentrate more on the upper body, so long as that's not a permanent thing.
How to start progressing again
Here's what I'd recommend:
Pushups are good, but stop focusing on increasing the number of repetitions, and find a way to add some weight to it. Rather than doing 20 reps per set, find a weight that limits you to 8-10 reps per set. It could be in the form of someone placing some light weight on your back, or you could find a weighted vest that allows you to adjust the weight.
Definitely continue trying to do pullups or chinups. If you focus only on pushing exercises, and too little on pulling exercises, we're once again concerned about muscular imbalances, this time in the shoulder/neck region.
Add at least one exercise for your legs to your routine. I would suggest reverse lunges, and also here, if you find yourself being able to do dozens of repetitions, start challenging yourself to holding some weight in your hands.
I suppose there is a point to be made about seeking progress in an advanced age, but 62 isn't too old to get in shape and look good. But keep in mind that any and all pain should be scrutinized, preferably with the consultation of a doctor or a physical therapist. That goes for all ages, but it becomes more important with every birthday.