There does not exist a polynomial upper bound.
Let $P_n$ be the number of partial orders on $n$ elements. It is know that $P_n \geq 2^{n^2/4}$. Thus, any method of uniquely representing the partial orders on $n$ elements, say in binary, will require at least $\log_2(2^{n^2/4}) = O(n^2)$ bits.
Now assume that for every $n$ there is a partial order on $n^k$, or fewer, elements, where $k$ is a constant, that is universal for the class of partial orders on $n$ elements. Fix some canonical ordering of the partial orders and let $U(n)$ be the first universal partial orders on $n^k$, or fewer elements.
Label each of the elements in $U(n)$ with a unique number from $1$ up to $\log_2(f(n)) = O(\log n)$ in some fixed canonical way. Now each partial order on $n$ elements can be uniquely described by writing down for each element that elements corresponding label in $U(n)$. This takes $O(n\log n)$ bits. However; this representation is not quite complete, as it seems to require the description of $U(n)$ to actually reconstruct a partial order given its representation in this form.
However, since $U(n)$ is the first universal partial order on $n^k$ or fewer elements, rather than appending an encoding of $U(n)$ to each partial order directly we can instead append an encoding of the following Turing machine $M$. $M$ takes in three arguments $n$, $i$ and $j$ and accepts if element $i$ is less than element $j$ in $U(n)$ and rejects otherwise. Given such a Turing machine we can clearly reconstruct the partial order. $M$ simply enumerates all partial orders of size between $n$ and $n^k$ and stops at the first partial order that is universal for all partial orders on $n$ elements. It then labels the elements of $U(n)$ in the canonical manner and accepts if the element labeled $i$ in $U(n)$ is less than the element labeled $j$ in $U(n)$. This TM has constant size.
We can thus uniquely and completely represent all partial orders on $n$ elements by $O(n\log n) + O(1) = O(n\log n)$ bits, which is a contradiction as there are too many partial orders on $n$ elements to be represented in only $O(n\log n)$ bits.
