Maybe. I've never seen a lease prepared that way, and would want to see the whole document and hear the rest of the circumstances before giving any opinion. It's possible to offer a lease, and have the offer expire January 1 - which would mean that if the lease was signed after that, the other party would not have to accept it. It's also possible that the restriction was waived. and the lease was accepted as is, even though it was signed "late".
As you state the facts, the contract was styled as an offer with a deadline date for acceptance of January 1, 2012. If you accepted after that date, you did not meet all the conditions of the offer, and so did not accept the offer as written. If the landlord or property manager thereafter countersigned the lease, then they would have ratified the lease contract making it binding. If they didn't, and only one party signed (you), and the lease would not binding.
However, a written contract can be accepted by conduct as well. For example, if you signed, and the landlord or manager allowed you to move in and never got around to signing the lease, the lease could still be binding on both parties (provided that the term did not exceed one year).
I really need to know more of the facts to be able to provide you with a better assessment. For example, did you move in? Your dates list 2012. So, is this problem a year old or did you mean to type 2013?