Yes, you could see plenty in that time.
Liverpool is a small city and everything is is walking distance, no need to hire a taxi for most things. I’d get one to go to Anfield, but you won’t need one to get around the city centre.
Walker art gallery, Liverpool Central library and natural history museum are all right next to each other. You could visit all 3 in an afternoon easily.
On the docks you have the Tate, museum of Liverpool, maritime museum, beatles story and you can do the ferry ride. You’d be pushed for time to do all of that in one day, so I’d probably advise picking aa few. The Maritime museum is the best of the lot, imo.
If you’re interested in The Beatles you’ll also want to go to the Cavern club & pub, and maybe also do the Magical mystery tour.
I’d guess you want to see the cathedrals, too?
The bus through the tunnel to birkenhead will take you 10 minutes each way. Wouldn’t advise spending too long in Birkenhead. 5 minutes is more than enough.
Most hotels in the city centre will be fine. Avoid the Adelphi. If you post any specific ones you find I’d be happy to have a look.