Well, I do not think it is the lid switch, but the next time it does it instead of advancing the timer press down on the lid, and if that does not start it up, lift up the lid and gently close it, again pressing hard. If this has no effect then the lid switch is not likely the problem.

Unless you plan to be in the apartment a good long time I would talk to the landlord and tell him, "look, when I move out you have to fix this to rent the apartment, so why not fix it now and maybe I will not move out as soon as I otherwise would"

It is going to be expensive to get a tech out to fix this, and you are not likely to get a quote over the phone (although you can ask how much to replace a timer, which will give you an indication of how much it might cost. Prepare for a big number)

If the landlord refuses to fix it I would just live with the problem, since you can make the machine work, and when you leave, well, then it is no longer your problem. Your landlord is not living up to his obligations even if they are not in the lease (check the lease and see if anything is in there that obligates the landlord to make the repair) he has a moral obligation to make the machine work correctly since you rented the apartment under the assumption you would have a working washer/dryer unit.

Ok, good. I feel better now that I know you new upfront that the unit was not covered in your rental agreement.

That is not a bad price to replace the timer. If you want to take a stab at it the timer sells for about $120 on this site, but you run the risk that it might be something else. Anyway I think you idea to live with it is the right one, and I assume you will not be staying there forever and it seems only slightly annoying to operate it this way. Also, it is one of those things that over time may even fix itself.

Thanks for updating us.