Fitness should be fun and varied. ITs by doing a variety of strength and balance exercises that we triain our bodies to develop control and strength in many different positions. The handstand is a very common exercise in gymnastics and gymnasts have very strong upper bodies and terrific balance so why wouldn't you copy what they do. It really targets the muscles in your back around the scapula and shoulders. developing strength in these muscles will improve your posture and the way you hold yourself physically. Use a wall to practise the handstand for safety and even a soft matt if it helps you to feel safer. I do handstands up against a wall because then you can't fall backwards. The handstand also develops excellent core strength and it is very important that you tense your tummy muscles and make sure that your back does not arch, try to keep a neutral spine as if you were standing!
Don't be afraid to give it a go and don't be afraid to suck. Practise makes perfect and the benefits are yours to earn. Start with minimal bouts of time and work your way up. You will develop good wrist flexibility and strength for exercises where you a pushing a weighted barbell such as the clean and jerk, bench press and the likes.
Check out some other upper body strength exercises The front dip and The Push up