Arm strength is less of an issue than core strength. As other posters said even with aggressive positions you shouldn't really be supporting weight on your arms. Some people will come over to road bikes from cruisers or more relaxed style bikes and just won't be able to support their upper body at really any angle, if that's the case, crunches and sit-ups are your friend as well as just general fitness. If your saddle to bar drop is too big or if your saddle is positioned with the nose pointing down, you likely just need to change you set-up.

I bet a lot of people that run super deep track drops have this problem and just never realize it because they don't ride on that bike long enough to kill their shoulders and back.

Cycling is a pretty bad strength training tool for anything but your legs.