You've noticed he can still reach the top for air so that's good.
The rest of my answer is assuming he didn't already have the missing hand when you bought him.
If he already had the missing hand he's possibly healed and fine.
If it's a new injury and not from a fish biting it off he might've gotten his hand stuck in the filter intake grid and in the struggle to free himself lost part of his limb. If that's a possibility you'll need to either change the filter to one that has a smaller grid on the water intake tube or put some mesh around the grid of the one you have like a nylon stocking (rinsed and cleaned thoroughly to clean it of dyes and chemicals), to prevent this from happening again.
On your Q if he will be ok - fungus and/or bacterial infection can set in on an open wound, esp. if they're living in under ideal conditions: a bowl of water.
A cycled, heated tank, very clean water and regular gravel vacuum are a must to prevent infection. If you see any infection starting you can treat the water with Maracyn Oxy. Use as directed for fish. Maracyn Oxy won't kill off the good bacteria in your filter. You can treat the water with both frogs in it, the other frog will be fine. You can use an anti fungal medication as a preventative too.
You can also use antibiotics like tetracycline, ampicillin, Maracyn Plus. ADF's can get secondary infections just like fish can.
The hand won't grow back on an adf like they can on an acf but adf amputees live long and happy lives (5-8 years) as long as they can get air and live in a cycled, maintained tank just like for fish.