Most apartment hunters find either July or November to be the optimal times to look for an apartment. A good tip is to start a week before the month in question, that way you'll notice any apartments where renters happened to give notice a little earlier than they were required to.
To make a good impression on landlords in Brownsville, ask them what the requirements of an application are before you show up to see the apartment. That way, if you're interested in the place you can fill it out immediately and have all the necessary supporting documentation handy. This is a great way to show the landlords you take your apartment search seriously.
If you're travelling Monday through Saturday, the Brownsville Urban System bus routes should provide you with all the connections that you need to see the city. The first route starts just before 6AM, and the last drop off is just before 8:30. If you need to travel outside of those hours or on Sundays, you will want to consider having a car in Brownsville.
Brownsville is a small town that is rich in culture, with Fine Arts, Historic and Children's museums, all of which are worth a visit. State parks nearby such as Resaca de La Palma and Boca Chica provide great outdoor recreation options. To hang out with the locals for a beer, try Shenanigans for an authentic Irish pub, or the Chicago Nightclub for an authentic Tejano experience.