I agree with the poster that would clean out the entire contents of the fridge on Friday.  I would post a note Wednesday of my intent with a time on Friday to have contents removed or all will be destroyed.  As for cleaning the fridge that part I probably would not do - removing old containers yes.
My current position was open for several months before I came along - I had to do deep cleaning in my area, inlcuding contacting the janitorial staff to do a better job.  (That incuded complaining about the rest of the office's cleanliness which still isn't up to standard in my opinion.)
Short answer I would 'empty" the fridge the first Friday, then speak to my boss that I did this as a one time good deal, however I'm sure the company does not want to pay my hourly rate for maid service.
And finally I would suggest as one company I worked for did - drew up a calendar for monthly break room duty between departments.  It was up to the department manager to assign the duty.  This company was big enough that a department would only have breakroom duty once every ten months or so.