If this is something you actually want to change, a few ideas beyond showers:
1) bike slowly (mentioned below; some folks don't like it, but the "slow" speeds up over time. I keep my HR under 140 and don't really sweat unless it's very hot, and this does still improve cardiovascular base fitness)
2) wipes (baby wipes, deodorant wipes, whatever)
3) clean shirt in bag
4) use a pannier, not a backpack
5) if you biked really hard, 5 minutes before arrival slow down and let your heart rate calm and let the breeze dry you off
Yeah, reason number one why I would never bike to work.
Even if I changed all my closed I doubt I would feel clean.
More and more places of work have showers available. Quite useful for biking in. Alternatively, you could go at a leisurely pace, but that's not as fun / makes it harder to justify the time spent.
Do either of those matter more than the exercise and cognitive improvement?
Well having a quality social life is imperative to mental health
Pro tip: dry shirt in your backpack.
Excercise is great but it does matter a little if i would be sticky and smelly.