Depending on the data that you have to work with, I'm assuming that you have point files for bus stops and housing projects, and a line file for the sidewalks, you could accomplish the task with a select by location, a buffer, and another select by location.
Thanks for responding! I'm not sure I understand you. I need to basically draw a line from the housing project to the bus stop using the sidewalk network. This is related to a concern that there are gaps in the network so that even though a development is within a 1/4 mi. buffer from the bus stop (as the crow flies), it isn't actually accessible because it takes considerably longer to get there by using the street network, there are no sidewalks, etc.
The other route to go if you do not have access to Network Analyst would be to identify all the bus stops that are within a distance of a sidewalk using select by location, then do another select by location based on the previous selection to identify the housing projects within 1/4 mile. Though with this route you'd have to then spot check the final results to determine the gaps in the the sidewalk network.
If you have your own sidewalk data, you will probably want to use it to create a network dataset. A network dataset is required as input for all the Network Analyst tool, and it contains the street (or sidewalk) network as well as properties that determine the travel time or distance across each network edge. After creating your network dataset, you will probably want to use the Closest Facility tool to find the bus stop closest to each housing project and the travel time between them.
If you have GTFS public transit data (the kind used in Google Maps), you can actually do a slightly more nuanced analysis using the BetterBusBuffers tool, which you can download from ArcGIS Online here: http://www.arcgis.com/home/item.html?id=42e57c5ff9a0497f831f4fced087b9b0. This tool can count the number of transit trips available at a set of point (your housing projects) during a time window. So, instead of just measuring the distance to the closest stop, it can actually count how many buses use that stop, which gives you a more complete picture of how much transit access the housing project residents have.