Modules and epimorphisms Let M be a finitely generated R-module ,where R is a PID and N a submodule.Is there an epimorphism from M to N?
 A: Yes, but the epimorphism might not doing anything nice to N ≤ M.  Just use the invariants.  What do the invariants of submodules look like?  What do the invariants of quotient modules look like?
For instance R=Z, M=ZxZxZ/4Z has invariants (0,0,4) and its submodules have invariants (0,0,4), (0,0,2), (0,0), (0,4), (0,2), (0), (4), (2), and ().  Its quotient modules have many invariants, but amongst them are all of the submodules' invariants.
You could also handle the free and torsion parts separately to get a cleaner proof.  The submodules of the free summand are easy to understand, being free of at most the same rank, and clearly each is isomorphic to a quotient module.  The submodule lattice and quotient module lattice of the torsion summand are isomorphic, so you can do better than just "some random epimorphism".

Invariants of submodules of (bounded) torsion modules: If R is a PID and M is a torsion module, then M is the direct sum of its submodules Mp = { m in M : there is an n such that pn⋅m = 0 } where p varies over the set of (nonzero) primes.  If N is a submodule of M, then Np = N∩Mp by definition.  To study submodules of torsion modules, it suffices to study the submodules of their p-parts, Mp.
If Mp is finitely generated, then there is a largest n needed to get pn⋅m = 0.  In other words, we are looking at modules M such that pnM = 0, for some prime p in the PID R (or any noetherian, commutative ring where p is a principal maximal ideal).  Such an M decomposes into a direct sum M = (R/pR)(r1) ⊕ (R/p2R)(r2) ⊕⋯⊕ (R/pnR)(rn).  Call such a module (p; r1, r2, …, rn).
Details: Two such modules are isomorphic iff after removing any "trailing 0s" (that is, with rn ≠ 0) the labels are identical.  Any such labeling represents a bounded module as long as the ri are all cardinal numbers, and the module is finitely generated iff the ri are all finite.
The isomorphism classes of its submodules are precisely the (p; s1, s2, …, sn) with 0 ≤ si + si+1 + … sn ≤ ri + ri+1 + … rn.
For example Z/9Z × Z/3Z × Z/3Z has label (3;2,1) and its submodules have labels (3;0,0), (3;0,1), (3;1,0), (3;1,1), (3;2,0), (3;2,1), and (3;3,0).  The submodule Z/9Z × Z/3Z × 1 for instance has label (3;1,1), the submodule Z/9Z × 1 × 1 has label (3;0,1), and the submodule 3Z/9Z × Z/3Z × Z/3Z has label (3;3,0).
