If a two-tailed test shows significance and the mean is higher, can you infer that one condition is significantly higher?
Yes.

A two-tailed test simply corrects for the appropriate probability of observing a difference as large as you observe given the null hypothesis is true. This probability is twice the one-tailed probability (in effect, a two-tailed test with alpha=0.05 is the same as two one-tailed tests each with alpha=0.025). You do not need a one-tailed test to infer the direction. If there is a significant difference with a two-tailed test, it clearly must be in the direction of the observed difference.