Zinc and copper salt bridge replacement
No, it would not.

$\ce{Al}$ is normally covered by oxide layer which effectively renders it inert to most things. Your cell would stop dead, as if the electrodes were made out of wood.

Then again, you might want to know whether it is possible to remove the said layer. Yes, it can be done, but then $\ce{Al}$ turns out to be just too active. It reacts with water quite readily (albeit not as spectacularly as $\ce{Na}$). Merry little bubbles of $\ce{H2}$ will pop up, flakes of $\ce{Al(OH)3}$ will drop to the bottom, and in a matter of an hour or so your electrode will be gone. While it lasts, the cell will produce some voltage; as Dr.J. correctly noted, it will be even greater than that of the normal $\ce{Cu/Zn}$ cell.

So it goes.