I think you will be OK. It would be even better if you used a higher quality plywood with more layers.
I have built birch plywood cabinets and didn't want the end grain to show, so I made 3/4 X 3/4" strips of solid soft maple (has a similar grain and color of birch), then covered the plywood end grain with these strips. I used biscuits and glue to attach them and then a flush trim router bit to trim them even with the plywood surface. Doing this not only covered the plywood end grain, but also strengthened the plywood edges and helped prevent the plywood from warping, kind of like a breadboard edge on a table, so it might be worth doing. Cross grain gluing isn't a problem when doing this, because of the stability of plywood. I have never had one of these joints fail.
The A Grado B means that one side is A grade and the opposite side is B grade. A grade is a smooth unblemished surface. B grade can have minor blemishes, but the surface is smooth. Sometimes they patch knots and blemishes to make the surface smooth. This is acceptable for a grade B surface, but not for a grade A surface.
If you bought this plywood from a big box store, the surface veneers are likely paper thin, so do as little sanding as possible or you might sand through the surface veneer. The higher the quality of the plywood, the more internal layers it will have and the flatter and more stable it will be. The plywood called "Baltic Birch" will have many thin layers and no internal voids in any of the layers, so it is more stable than plywood with fewer layers and internal voids, but true Baltic Birch plywood is metric dimensioned and about a 60" X 60" sheet size.
Charley
