This is a familiar redox reaction, occurring through the intermediacy of hydrogen molecules. As expected, it can be expressed in terms of two ionic half-reactions; one reduction half-reaction and the other oxidation half-reaction.
The reduction half-reaction is: 2NO2⁻ + 7H2 + 12e⁻→ 2NH3 + 4H2O
And the oxidation half-reaction is: Al + 2OH⁻ → AlO2⁻ + H2 + 3e⁻
These two ionic half-reactions add up to give the final ionic reaction.
The reaction can then be balanced easily by cross-multiplying above two half-reactions for electron balance followed by minor adjustments aimed at balancing other atoms.
1). 3e⁻+ 2H2O+ Al → AlO2⁻ + 4H⁺ ⇒ ❶
2e⁻ + 3H⁺+ OH⁻ → H₂+ H2O ⇒ ❷
2). ❶ x 2 + ❷ x 3 ⇒
H⁺ + 3 OH⁻ + 2Al + H2O → 2AlO2⁻ + 3H₂
2 OH⁻ + 2Al + 2H2O → 2AlO2⁻ + 3H₂