A tenant can be charged for the last (final) month's rent at the beginning of the tenancy. The lease is a contract between the landlord and tenant. If the tenant objects to this provision in the lease or any other provision(s) in the lease, the tenant should not sign the lease. There are many landlords who do not charge last month's rent at the outset of the tenancy. It is the landlord's discretion whether or not to charge last month's rent at the beginning of the tenancy. Some landlords may decide to charge this to insure that the tenant won't just move out at the end of the tenancy without paying the final month's rent. If paying last month's rent at the beginning of the tenancy is a financial problem, it would be advisable to find a place that does not require advance payment; especially when considering that the tenant will most likely be paying first month's rent and a security deposit prior to moving into the apartment.
The security deposit is not the same as last month's rent. The security deposit is for cleaning when the apartment is vacated by the tenant at the end of the tenancy. If you really want the apartment where last month's rent is being charged, you might want to try to negotiate with the landlord perhaps to pay that charge in installments.
With regard to the noisy neighbor in the apartment building, every lease includes a covenant of quiet enjoyment which means that the tenant cannot be disturbed in his or her use and enjoyment of the premises. Although the tenant could file a lawsuit against the landlord for breach of the covenant of quiet enjoyment, this should only be as a last resort. First, complain to the landlord or the landlord's agent such as the property manager regarding the noisy tenant. Reduce your complaint to writing so that there is a record of it in the event of future potential legal disputes. The landlord should take action against the noisy tenant. If the noisy tenant does not comply, the burden is on the landlord to evict the noisy tenant. Keep in mind that unfortunately some noise in a neighboring apartment may be unavoidable due to thin walls, including a telephone ringing, conversations, or perhaps someone walking upstairs or outside your apartment. The issue is what constitutes excessive noise? Excessive noise would be blasting a stereo or TV or noise late at night violating quiet hours. Check your lease for provisions designating quiet hours. A violation of quiet hours would strengthen your case for a breach of the covenant of quiet enjoyment.
Before you terminate your lease for a breach of this provision, contact an attorney in your state that specializes in landlord-tenant law. Terminating prematurely, incorrectly, or under the wrong circumstances could result in a forfeiture of your remedies or deposits.