2-tape TM that accepts words from $L := \{bin(n)$#$w \in \Sigma^* | n \in \mathbb N \land w \in \{0,1\}^* \land |w| \le n\}$ in $O(n)$
Think over how to convert binary representation of $n$ into unary one using two tapes. The result should be stored on the second tape. For example, if $n=6$ then the unary representation could be $111111$. Then compare number of symbols of $w$ and number of $1$s of the unary representation of $n$. If the number of $1$s is less than or equal to $|w|$ then accept, otherwise reject.

Since number of bits in $bin(n)$ is no more than $log(n)$ number of unary bits is no more than $2^{log(n)} = n$, so writing down each bit on the tape should not take more than $O(n)$ time, and comparison $w$ with the unary $1$s also takes $O(n)$.

So you just need to design and algorithm for conversion from binary to unary representation of $bin(n)$ using two tapes that runs in $O(n)$ time.