Here is the little I could glean from the literature about the actual tasks of the cellarius. Celarii are mentioned frequently enough in texts but there is very little about their tasks, unfortunately.
- dispensing wine
- heating up the wine
- flavouring wine
(for all of the above see: Plautus, Miles Gloriosus, act 3, scene 2)
- doctoring wine to hide disagreeable odours (see: Pliny, Natural History, 19.67.188)
In Plautus' Captivi, Hegio makes Ergasilus his cellarius (act 4, scene 3). Ergasilus is ecstatic about this; it means free food! He makes a little speech that tells us that the cellarius:
- was in charge of the food (illic hinc abiit, mihi rem summam credidit cibarium / he's gone away and left me in charge of the food)
- prepared the food to some extent although not cooking as such (see below) and this doesn't include butchering as Ergasilus hopes to catch them napping (so he can eat in secret)
- inspected the food
- helped in tasks such as hanging the ham
In Martial 11.31, the cellarius seems to be preparing some sort of hors d'oeuvre - capelliana in rue leaves.
Not the cook or other household slave
As noted above, Ergasilus mentions butchers specifically, indicating that they are a separate type of servant.
Cellarii are also distinguished elsewhere from cooks, pastry chefs, and household servants generally (see: Seneca, Epistles, 122.16; Columella, On Agriculture, 12.2)
It's not much to go on, I'm afraid! I include a link here to William Smith's entry on cella, which gives more information on the sort of things kept in cellars which may pad out our picture of the cellarius' job a little.