I went back and looked at the schem for the tape echo circuit, it has a supply splitter as mentioned.
The op amp spec'd will work, so will others like an Lm324.
Dont get caught up too much in 'split supply' op amps, for your purposes its not so much an issue.
Nope you cannot take 4.5v from the voltage divider to the cassette deck, the divider cannot provide nearly enough current.
The easiest route would be to power the deck from a separate wall wart, its possible to run them both off one supply, probably using a regulator like a 7805, but you'd be inviting lots of other issues with Dc offsets & ground currents causing hum.