So, there's really two questions here....(1) is
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Your lease likely ended on March 31, 2013. New York landlords do not usually end a lease on the first day the apartment might be rented to a new tenant,
Unless your lease requires greater notice, a tenancy measured by a written lease ends on the last day of the lease without any need for any written notice. The lease ends on the last day.
You need to review your lease to see the actual last day of the lease. If as the landlord claimed it is April 30, 2013 then that is the day to vacate. If you stay behind, New York law defines your tenancy as a holdover tenancy. The landlord would be allowed to file a holdover petition on the first court date following April 30, 2013.
If you stay one day beyond the expiration date of the lease, New York law makes you liable for the entire month for that one day.
If you holdover, depending on terms that may be on your lease, your landlord may charge a penalty rent (use and occupancy) greater than the rent last paid under the lease.
Review your lease for all terms that may apply to the end of the lease, your obligation to move out, any advance notice that may be required to be provided to you, and for any penalty provision in case you become a holdover tenant.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.