Please, go here  to find all the options (four) and instructions on how to get to Alcalá de Henares from the Barjas-Adolfo Suarez airport (MAD).
Summary of getting to Alcalá via Bus (prefered option):
The recommended option is the bus 824 from Terminal T1 or T2 of the airport. From T3 you can walk to T2. From terminal T4 you need an internal bus between terminals. Check internal bus connections here.
Find here a map of the route of bus 824 from Barajas (Adolfo-Suarez) airport to Acalá de Henares-Brihuega stop.
From the bus stop (Alcalá-Brihuega) you need to go to the residence Lope de Vega (city center), following this map instruccions.
Summary of getting to Alcalá via Train (second option):
You can also go by train (it is call Cercanías and the symbol is a letter "C"). See the network map here.
You need to take the train in terminal T4 (if you are in other terminal, check the option before on how to change between terminals). The direction of the train must be in the direction of Principe Pío and change the train in Atocha (you can also change stations sooner) looking for C2 or C7 line in the direction of Alcalá de Henares or Guadalajara.
You need to get off at Alcalá main train station and from there to the Lope de Vega residence (city center): map here.