Both those phrasal verbs work if you're thinking of the person as one of a team/set... If you mean 'take over for me as boss', you could say 'mind the shop' (it doesn't have to be a real shop), 'hold the fort', or 'take [over] the reins.
You only need to add 'at work' if that's not clear from the context (which, for a work-mate, it probably is.)
'Fill in for me with X' doesn't work unless X is a person. In those circumstances you could say ' Can you take over my paper round on Sunday?' or '...fill in for me on my Sunday paper round?'