Since there are no classes I'd think that all university offices would be closed all week of Semana Santa.

There are processions in Madrid during Semana Santa but much much smaller as compared to those in Andalucía.

Nightlife during this time in Madrid is .... subdued, I'd say is the right word. It seems that about 33% of the city leaves during this week but since no one is working many many people are patronizing the restaurants and bars most days/nights.

In many regards, visiting Madrid during Semana Santa is the BEST time of the year to visit because the streets are quieter and there's MUCH less traffic. It might even be the BEST time for your first visit.

Enjoy your time in Spain. Saludos, MadridMan