If a mezuzah falls while in its case on Shabbos, is it able to be handled regularly or it must be picked up in a irregular manner since it is muktza (since while in its case it does not serve any use)? Source please.
A Mezuzah that will be put back on the door is Muktzeh on Shabbos.
However, it is permitted to move it from the floor so that it is not in a state of bizayon, and as you write it is best to handle it in an irregular manner.
See Minchas Shabbos no. 88; Shut Az Nidberu 8:38; Shut Shraga HaMeir 6:9; although some write that a Mezuzah is not Muktzeh (see Shalmei Yehuda p. 28; yet, it seems that people are makpid about Mezuzos and won’t use them for study etc.). For a Mezuzah that fell see Shemiras Shabbos Chap. 20 note 33.