Getting Started →
Do I need a lot of upper body strength?
The answer is: Not initially, but at some point... yes.
But the answer doesn't matter. A lot of < 5.10 climbing is a test of your ability to balance your body, keep your weight over your feet, and move efficiently. Your arms are primarily working to keep you from falling back. They shouldn't be doing much to propel you up.
So if you can't do a pull-up, it's really not a problem.
In fact, because most beginners haven't developed any climbing-specific muscles, it's actually remarkable how similarly most people start out — regardless of body composition or genetics — and how predictably they progress.
As you do progress into the harder grades, you'll find that strength in your forearms, fingers, shoulders, core, and back develops along with your understanding of climbing, your awareness on the rock, your confidence, and so forth. So you'll get better at climbing just by climbing. And, in that sense, the upper body strength will appear "for free" and you might not even notice it.