The only way to support schedule-based public transit in Network Analyst is to download the Add GTFS to a Network Dataset tool: Add GTFS to a Network Dataset | GTFS in ArcGIS
To accomplish what you're trying to do, you would have to use this tool to create a network dataset that includes the transit schedules, then publish your service based off this network dataset. You also have to be able to install the "transit evaluator" on the server machine. There are some instructions within the tool's user's guide that talk about that.
Note that you can get the closest facility and the travel time, but you cannot get directions that are sensible for public transit.
