Yeah seriously...dips? Why not tell somebody to improve their running by riding a bike while you're at it? It's better than nothing, but dips don't work the same things.
Also, deltoids isn't what fails for me, my triceps burn out first. Just do them to failure at some point, and figure out what muscle group fails on your first. If you honestly feel like that muscle group is holding you back, then maybe work on it, but otherwise just doing a butt ton of pushups is your surest bet.
I have noticed deltoids help if you're worried about being able to stay in the pushup position, as would abs to stabilize. But on the PRT it's IMO better to just bust em out as fast as you can, then drop and take a rest. I did about 70 in like 1:15, which gave me almost an extra minute to rest.
So personally, it might seem like I'm being facetious/lazy when I say just pump out near max sets throughout the day, but I'm actually pretty serious.
You usually are able to do a quick set of reps, then you get tired, and the next batch is slower. Then you start needing more rest between batches, so you get maybe 3-4 pushups every 10 seconds.
So by blasting out a whole lot of near max sets throughout the day, you basically are focusing on increasing the number you can pump out quickly to start with. So I basically knocked out my good low in way under a minute, took a little break, then knocked whatever more I felt like doing.
I also like to go lower and slower during those sets than I regs require during practice. That way, when you only have to break 90 and get go as fast as you like, it's unbelievably easy.