Can Eve perform this operation?

Please be careful with your notation; don't get confused with the number of qubits that are in $|0_{Eve}rangle$ (I'm not necessarily saying that you are - just mentioning this as a precaution). Since the state $|psirangle = frac{1}{sqrt{N}}sum_{j in 0,1,2....N-1}|jrangle$ that Alice wants to send is a $n$-qubit state with $N=2^{n}$, Alice needs to send $n$ qubits to Bob - possibly via Eve. Therefore, the state $|0_{Eve}rangle$ that you describe is a $n$-qubit state as well.

A unitary $U$ that copies the state $|psirangle$:

The $2n$-qubit unitary that you describe is possible, but on the presumption that Eve knows exactly what Alice will send. Then, a series of $n$ $CX$ gates from the $i$th qubit to the $(i+n)$th qubit will suffice - it will copy any state $|jrangle$ in the first $n$ qubit to the second $n$ qubits, thereby obtaining the total state $|jrangle otimes |jrangle$. By linearity, it also copies a superposition of different $|jrangle$-states. So we have: begin{equation} U_{copy} = sum_{iin {1,2...n}}CX^{(i,i+n)}, end{equation}

where $CX^{(i,i+n)}$ is the CNOT operation with qubit $i$ as the control qubit and qubit $(i+n)$ as the target qubit.

A unitary that copies any state $|phirangle$:

However, if Eve does not know what state Alice sends, there is no unitary that will perform the task (of copying a unknown $n$-qubit state $|phirangle$ into the second $n$ qubits). The proof of impossibility is straightforward via the condition that the operation needs to be linear; for a proof see a proof of the no cloning theorem, which applies here.