Certainly you are not required to accept a unit that is not fit. Hopefully there is no provision in your rental agreement that states that you inspected the unit and found it suitable before you executed the agreement. This would complicate things perhaps, but whether or not there is such a provision, I cannot see any judge ruling that you had to accept the apartment in unclean and damaged condition. It is good that you alerted the landlord about your complaints before you took possession. Unfortunately because of his assurance that he will correct the problems before you do take possession, I think you now are bound to allow him to deliver on his promise.
Practically speaking, though, I have serious doubts about this guy performing reasonably. I think you are taking a big risk here. Almost all landlords know that they have to get their apartments rent ready even before they show the units, much less before the tenants takes possession. You may be stuck for a couple of weeks, but I would be on guard against a slipshod job by a landlord who appears deplorably ignorant or cavalier. If the apartment is not reasonably clean condition and good repair on the day you are scheduled to move in, I recommend you rescind the agreement unequivocally and immediately. You are not required to accept substandard living conditions.
Under the law of contracts, you have a right to demand assurances when you reasonably believe that the other party is not going to be able to perform. You may want to talk to an attorney about invoking that rule here. Some risk is involved in that your belief of nonperformance may not be reasonable. But you don't want to learn on move-in day that the unit is still uninhabitable.