How to convert sodium aluminate back to sodium hydroxide?
I don't think you will have much luck recovering the hydroxide.  To do so you would need to reverse the reaction that happened when the aluminum dissolved, which was

$$\ce{2 Al + 2 NaOH + 2 H2O -&gt; 2 NaAlO2 + 3 H2}$$

Your idea is essentially to substitute reduction at an electrode for reduction with hydrogen gas.  Unfortunately electrochemical reduction of $\ce{Al(III)}$ is very difficult.  If your idea was feasible, it would revolutionize aluminum manufacture!  Instead of doing electrolysis of molten cryolite at ~1000 degress with sacraficial carbon electrodes, aluminum manufacturers could use your process.  So if you could get it to work, you'd be rich...but I am not hopeful.

Note however that you don't necessarily need to do electrolysis to recover sodium hydroxide.    There is hope!  You could just boil down your solution so that it is supersatured with respect to aluminum hydroxide, and then wait for the aluminum hydroxide to precipitate, which would happen faster if you add seed crystals of aluminum hydroxide $\ce{Al(OH)3}$) to initiate precipitation.  

$$\ce{2H2O + NaAlO2_{(aq)} -&gt; Al(OH)3_{(s)} + NaOH_{(aq)}}$$

This step is part of the Bayer process for commercial refining of aluminum ore.