The astral battlefield is a second battlefield otherwise identical to the regular battlefield, except that it doesn't exist unless a player controls an astral permanent. Permanents can't interact with each other across battlefields, and static effects don't reach between the two. Spells with multiple targets can't target permanents on separate battlefields, but permanents on the astral battlefield can be targeted as normal otherwise. When casting spells like Overrun, you choose which battlefield your spell effects as you cast it.

Astral cards would be primarily blue and black, with some white and red and few green cards.

Is this a bad idea? Probably, but I can't just let it fade from my mind.

if colours don't have a lot of astral representation, then they need other ways to interact with that battlefield

provided they have that - and assuming you're designing a self-contained environment - seems like a cool idea to me. might need some differentiation to avoid seeming like shadow 2.0, but there's certainly scope for that given it's a supertype rather than a keyword ability

Since only permanents have the astral supertype and you choose which battlefield nonpermanents affect, a spell that creates tokens can place those tokens on the astral battlefield, so long as there's already an existing astral permanent. So token creation spells would exist to allow flexibility in that regard.

I imagine green could be the anti-astral color, and could get "destroy target astral permanent" just as it gets "destroy target creature with flying."

astral reach?

I know that wouldn't be the actual name, but it sure sounds cool.

Actually, maybe red could be the anti-astral color. Either could be flavorful.