Pull-ups are showy but don't mean much to a cyclist. The only time I really use my biceps is sprinting hard OOS. Pushups are what's good for cycling. Works your chest, arms, and core. Being able to do 20 strict ones is enough for most of us.

That said, being able to hang from a bar for a minute is a very good thing. Hanging opens the space in our shoulders. Years ago, I was having shoulder pain. I got a prescription for PT but the PT did absolutely nothing for said pain. I searched around and found this book:
Shoulder Pain? The Solution & Prevention, Revised & Expanded: Kirsch M D, John M: 9781589096424: Amazon.com: Books

Fixed me up in just a few days. It's a bit of an odd thing: what apparatus do grade-school kids gravitate to? Monkey bars. That's because we evolved from folks who did a lot hanging from their hands, and we still have those same shoulder structures and that same desire to hang, though most of us grow out of it.

One minute/day is all it takes. If you're doing chin-ups, just hang, fully relaxed, for a few seconds between reps.