You sure can do it, nothing wrong with this. Being a used mower and not knowing how it was cared for, I'd run some cleaner too.
My OPE is a mix of old and new stuff. I keep a single 5 gallons gas can for all this (except of course 2 strokes), this fuel receives MMO treatment.
As for the oil, I never really tried any additives in there. I once added MMO to my oldest car's oil, but in much smaller dose than recommended so it may not have done anything.