It's not hard to find upper and lower bounds within a constant factor of each other. In one direction, the sequence k, 2k, 3k, ... k−1, 2k−1, 3k−1, ..., k−2, 2k−2, 3k−2, ... etc of length k2 is universal. (This is the same tilted grid pattern that gives the lower bound in the Erdős–Szekeres theorem.) In the other direction, the length n of the universal permutation must be at least some constant times k2, in order for it to have at least k! different subsets of size k. But what is the constant?
A computer search found that there is no 4-universal permutation on 8 items (the best is 51862473 which has 23 out of the 24 possible patterns), but the 9-item permutation 162975384 is universal. So the sequence of lengths of the shortest k-universal patterns begins 1, 3, 5, 9, ... But although there are simple quadratically-growing sequences like ceiling((k^2+1)/2) that begin like this, it's not really long enough to identify this in OEIS and see whether it matches anything known.