The question is in the title:

Question: Is there any 4-dimensional polytope without 3-gonal and 4-gonal faces (of dimension two), other than the 120-cell?

I consider only convex $d$-polytopes $P$ (convex hull of finitely many points) that are full-dimensional (i.e. $\mathrm{aff}(P)=\smash{\Bbb R^d}$). And I consider a polytope to be distinct from the 120-cell if it has a non-isomorphic face-lattice.

It is known that any 4-polytope must have a 3-gonal, 4-gonal or 5-gonal face of dimension two (what is a source for this?). The 120-cell has only 5-gonal faces of dimension two.