I have to do a Mann-Whitney U test in SPSS and it has to be alpha = 0.01, one-tailed.
I know the default is 0.05, but I don't know how to change it to 0.01 in SPSS (for Mann-Whitney U test).
Is it even possible?

SPSS returns the two sided p value. Dividing it by 2 gives you the one sided p value (assuming that the direction of the difference between the two groups is in accordance with the one sided alternative hypothesis). You can just compare the one sided p value with the alpha of 0.01. If the one sided p value is equal to or smaller than 0.01, you can reject the null hypothesis.