Suppose $f$ is a weight $2$ level $N$ cusp form. When can we realize the mod$\ell$ representation of $f$ in a form of weight $2$ and level $Np^3$, where $p$ is some prime not dividing $N$? I assume that, if a simple criterion exists at all, it is a condition on the mod$\ell$ representation of $f$ restricted to inertia at $p$, but I'm not sure what it would say...

Presumably you want the form (let me call it g) of level Np^3 to be new at p, otherwise it's trivial. Let me also assume ell isn't p. If the form g is new at p, and has level Gamma0(p^3) at p, then the elladic representation attached to g will have conductor p^3. But this is a bit of a problem, because the conductor of the mod ell representation can't be that much lower than the conductor of the elladic representation. Indeed a theorem of Carayol and, independently, Livne, says that the pconductor of the mod ell representation will be at least p if the pconductor of the elladic representation is p^3 (the exponent can drop by at most 2). So if you're looking for Gamma_0(p^3) then you're in trouble. This is just a local calculation and isn't too deep. Diamond and Taylor, in their second paper on the subject, give a list of the conductors of the newforms that can give rise to a given irreducible modular mod ell representation. You can see that Gamma0(p^3) is too much from the main theorem there. Of course the work in that theorem is realising everything that is possible, not ruling out everything that isn't. 

