I love the Algarve, beautiful beaches and friendly people. Would recommend
http://www.vilapetra.com/UK/ we stay there quite a bit, at that time of year it's 63 euros a night for a studio (bonus is that it has a completely separate kitchen with dishwasher, it's not all in one room). It has well deserved good reviews on TripAdvisor. It's in a fairly busy area of Albufeira (some call it the new town, some Montechoro, some Sao Jaoa) so there are restaurants on your door step. Approx 15 min walks to the beach, and you can walk/taxi to the old town, or there's a new marina on the other side. There's a place called Zoomarine where you can swim with dophins nearby. We also got a villa one year through
http://www.holiday-rentals.co.uk/ and that may be within your budget if you can get some cheap flights. (Albufeira is about half hour taxi drive from Faro airport, probably looking at £30 each way, or could get shuttles cheap - have a look at
http://www.resorthoppa.com/ or
http://www.transfers4u.com/
Just found the villa we stayed in, it's not the most glamorous, it's quite traditional and is someone's home from home, but we loved that about it! The location was really good (near the Dom Pancho hotel) and we loved having a private pool. At that time of year it is £855, so again if you could get some cheap flights it would be within budget. If it's for this year and they're not booked I would be cheeky and ask for some last minute/honeymoon discount, if you could get 10% off and with Easyjet flights you could afford (or other villas too).
http://www.holiday-rentals.co.uk/Portugal/Algarve/holiday-villa-Oura/p16597.htm