Hostel/Albergues near Lisbon Cathedral

Hi, Dianne, Welcome to the forum. On the chance that you might not have yet figured out how to navigate around, you should take a look at the guides and other resources about the Caminho from Lisbon and from Porto. There are guides written by forum members, suggestions on breaking up long stages, etc. https://www.caminodesantiago.me/community/resources/categories/camino-portugues.40/.

As far as a place to stay near the Cathedral, I would recommend the Pensao Prata. It's a popular place for pilgrims, it is a family run, clean and basic place. Great location about 5 minutes walk from the cathedral.
http://www.pensaoprata.net/v1/.

There is no pilgrim albergue in Lisbon. But there are a lot of youth hostels in Lisbon, which is a very "happening place" these days. Some forum members enjoy the activity, but I would never stay with a bunch of college kids whose main activities typically involve the hours when I want to be sleeping. But to each his/her own. You can find lots of these dorm-room places on booking and other websites.

Not sure when you are starting out, but just wanted to also make sure you knew about the new (as of last May) pilgrim albergue in Alpriate, about 20 km from Lisbon cathedral right on the caminho. I don't know the exact opening date, but the Via Lusitana who runs it has mentioned early April as a possibility.