Hi Andrew,
There are a couple options here...it is possible that the academic institution may have an ArcGIS Online organizational account you can be added to, you could create a 30 day free trial, or purchase a Home Use license of ArcMap which I believe comes with an ArcGIS Online organizational entitlement.
Here is a link to the Home Use page:
ArcGIS for Home Use Program | ArcGIS for Desktop Advanced for Personal Use
Here is a link to the ArcGIS Online public vs. organizational accounts comparison:
ArcGIS Online | Comparison Table
Publishing a zipped shapefile to a feature service for use in an ArcGIS Online web map is not supported at the public level. The customization of a web application would have to be done by downloading and configuring the application template and possibly writing and add-in using the Javascript API.
Here is a link to the ArcGIS Online Resource Center:
Javascript API: