You really don't need to crosshatch the 2x2s. In the upper part over the pass through, I would do the 2x2s on 16" or less centers making sure they are well supported on the ends. Where you have seen the supports attached to the plywood lid is so the space is not covered with the wood. If a lip is added so they can be supported on the ends, that would be the way I'd do it. Even if you add doors or drawers, having access from the top would be a good thing. You might also look into replacing the 1/2" with 3/4". It is also best to have the plywood perpendicular to the supports.
Good luck.
