These are instructions for reaching the Historic Campus, provided by the Universidad de Alcalá.
Alcalá de Henares is only 25km far from the Adolfo Suárez Madrid–Barajas Airport so the fastest alternative to get to the historical center from there is by taxi, Uber, Cabify (Spanish version of Uber), or similar. The ride will take about 20 minutes and will cost 30 to 40€.
There is also a direct bus 824 from Terminals T1 and T2. It will take about 40-60 minutes and cost below 5€.
There are more detailed directions in this document from the Universidad de Alcalá.
Google Maps works very well to find public transportation options in Madrid and Alcalá.
(These are the options it gives to go from the airport to the historical center.)