This is a big question, so let's try to break it down and generate some ideas.
The first thing you need to do is filter out the students who aren't eligible for bus passes because they live within the 1.5-mile distance of the school (I presume you're looking at network distance rather than straight-line distance.). The best way to do this is to use an Origin Destination Cost Matrix with a distance cutoff. This tutorial may help: https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/od-cost-matrix-tutorial.htm
Next, I presume the students will be using scheduled public transit service rather than riding school buses. Assuming you have GTFS data for the transit system, you can incorporate these public transit schedules into a network dataset. Here's a tutorial that walks you through that process: https://pro.arcgis.com/en/pro-app/latest/help/analysis/networks/create-and-use-a-network-dataset-wit.... Once you have the network created, you can model travel by public transit.
Since students are presumably dismissed from school at a particular time of day, you can eliminate some of the challenge in transit travel time calculations by assuming a given starting time of day. You can calculate the routes of these students from school to home at that time of day using the Route solver. The challenge will be in extracting information about the actual transit lines taken because our solvers don't report this information. However, you can download the Transit Network Analysis Tools from here: https://github.com/Esri/public-transit-tools. The Copy Traversed Source Features With Transit tool will give you more detailed information about which transit lines got used for each route: https://github.com/Esri/public-transit-tools/blob/master/transit-network-analysis-tools/UsersGuide.m...
Hope this helps get you started.
