Whenever you're going traveling for something like a month or more, it makes sense to search for an apartment instead of booking a room in a hotel. Yes, it does make sense. However, in Albufeira and during off-season I wouldn't be so sure about it. I've looked all over internet for apartments and the best price I could find was 200 euros a month and that was an offer without a picture. Other offers I have seen have been from EUR 300. And when I was walking around in Albufeira, the best prices I could see on the walls were EUR 350 for T1 and EUR 480 for T2. So if we consider that during off season you can get a room for yourself in a hotel for under EUR 300/month and this price even includes maid service, you shouldn't really bother finding an apartment.
But during high season and for longer terms it makes sense. But be aware, the prices START at EUR 300 and often go high up.