The sights as they come from the factory take some getting used to, at least for me. I like the factory sights on the Henry H001 better as far as sight picture. They look just like regular pistol sights. Williams, Skinner, Ranger Point, and some others make aftermarket sights if you're so inclined. For a considerable increase in precision without going to a scope, I'd recommend some sort of peep/aperture sight. Try the factory sights first though. They're already installed and you may like them.
If I weren't such an aperture sight type of guy, I'd probably leave the factory sights in place as long as they matched what I wanted to shoot out of them. The Henry rifles will shoot Short, Long, Long Rifle, CB, Quiet, Standard Velocity, High Velocity, Hyper-Velocity, Interceptor, Velocitor, and 22 Shot. That kind of variety means you'll have to test your ammo with your sight of choice. My H001 is set for Aguila Super Extra High Velocity (1250 fps) and Standard Velocity (1100 fps). One notch difference on the rear sight between the two out to 50 yards.
Cleaning my H001 consists of a bore snake carefully pulled through the barrel two passes, then a damp Q-tip to swab out the open action. Then I wipe the whole thing down with an oil cloth and that's it. It's my fastest to clean rifle.