In the Unearthed Arcana version of the Way of the Astral Self, the monk's Astral Arms were treated as monk weapons.
Under that wording, the monk probably couldn't grapple/shove with them because weapon reach and the reach of the character's hand that could make the grapple/shove are different things.
That is, it's hard to see how you could grapple with the end of a polearm, for example (through a lenient DM might allow a shove).
However, in the official release, the wording was changed then from monk weapons to unarmed strikes:
You can use the spectral arms to make unarmed strikes.
When you make an unarmed strike with the arms on your turn, your reach for it is 5 feet greater than normal.
Under this revised wording, it seems like the monk could grapple/shove with its Astral Arms.
But can it?