The short answer is, yes, it is possible to find an apartment that fast. However, it really depends on how flexible you can be.
Although it can vary from city to city, there are really two avenues you can take:
By broker, I mean you walk into an apartment brokerage and say "I'm looking for a [studio/1-bedroom/townhouse] for [$1200/$2100/$4000] a month, what do you have available?" There are dozens of these in any major US city. They will then setup appointments for you to visit and view any apartments that meet your criteria. If there are currently tenants living in any of these apartments who are on their way out, the broker will need to give them at least 24 hours notice that they will be bringing someone by to view the place.
If you find a place you like, the broker will contact the landlord/owner, and will take care of putting together the lease and running any necessary credit checks, etc.
As far as fees go, it is not at all atypical to expect to pay First Month's Rent, Last Month's Rent, a Security Deposit, and a Broker's Fee.
First Month's Rent = One month's rent
Last Month's Rent = One month's rent
Security Deposit = Often one month's rent, refundable upon inspection and move out
Broker's Fee = Often one month's rent*
*A note about broker's fees. The current landlord/owner will often agree to pay 50% or all of the broker's fee, instead of leaving this to the new tenant. Even if they haven't agreed to do either of these things, it is a good first step in the negotiation to ask if they're willing to cover/split the fee as a way of decreasing your initial investment.
By non-broker, I'm referring to going at it alone. There are any number of apartment related website that you can presumably find on Google, but most Americans just use Craigslist (http://www.craigslist.com) - particularly for urban areas.
It's important to note that brokers also post listing on Craigslist. In fact, Craigslist has filters for "By Owner" apartments (therefore no broker's fee), "Broker - No Fee" apartments (by broker's, but where the owner is paying the fee), and "Broker - Full Fee" apartments (as laid out in the "Broker" section above)
Typically how this works is you email the owner (or broker, where appropriate) via Craigslist and state your interest in the apartment. You ask when it might be available for a viewing, and then wait for a response. Once you've seen the place and determined your interest, you can either take it for the asking price, or negotiate a lower monthly rate. It's OK to negotiate! Just don't be surprised if the landlord is unwilling to come down on their price.