If you arrive in Santiago in June, July or August, the Pilgrim Office is VERY crowded. Each morning a huge wave of pilgrims arrives, mostly off the
Camino Frances. They typically arrive starting at 09:30. This causes a 60 - 90 minute queue for getting Compostelas to form. Most all of these folks are intent on getting their Compostela and making the Pilgrim Mass at noon. Only the first arriving folks are likely to succeed.
During the summer months, the Pilgrim Office opens at 8:00. Those waiting at the opening time are likely in and out in 30 minutes or so. So, if you can't get there first thing in the morning, you should plan to make a full day of it, and stay overnight before departing, at least IMHO. Make reservations EARLY. Santiago is frequently full to the brim during the summer months.
Too many of these mid-morning arriving pilgrims think they can come to the office, get their Compostelas quickly, attend the noon Pilgrim Mass, then catch a bus, train or plane out of town. Big mistake! Besides, Santiago has so much to offer.
IMHO, you are really short changing yourself if you rush through the arrival activities and get out of town fast. Especially if walking on to Finisterre and Muxia, consider sleeping overnight and then pressing onward.
Consider an albergue or hostal on the West side of Santiago.
Rua Hortas is the direct route to the coast and Finisterre / Muxia. If you say on the west side and are leaving the following morning, you have marginally less distance to walk. There a quite a few lodging options in that direction.
My advice, based on long experience, is to choose ONE accomplishment on the arrival day. Either wait in line for your Compostela OR attend the noon Mass. It is near-impossible to do both in the first order, Compostela then Mass.
N.B. There is a second large surge of pilgrims after the Mass, at about 13:00. This surge usually causes the lines to grow to more than a two-hour wait...just sayin...
But, if you persist, the optimal way to accomplish attending the Mass, and get the Compostela the same day is to arrive in Santiago, place your rucksack in daily consignment (consigna) or at your night's lodgings, attend the Mass, then stand in line for your Compostela, then retrieve your rucksack. By then it will be about 15:00 or so.
There is a convenient consigna in the Campus Stellae shop at the top of the Plaza de Praterias stairs, immediately to the right, on the corner. It costs €2 for the day, until close of business.
Good alternatives for daily consigna include: the main post office (Correos) on rua Franco, and the Pilgrim House at #19 Rua Nova. €2 is the standard daily charge around town. At the Pilgrim House, this is a suggested donativo amount - JUST DO IT!
You can also place your bag into temporary consignment storage at the Pilgrim Office for the same amount. But, in my experience, and IMHO, the process to check the bag, and to retrieve it takes too long. If you are in a rush, this might prove a hindrance. Consider which consignment location best fits your day's activities.
I hope this helps.