Granada is one of the few cities in Spain where a tight budget and a genuinely good trip are not in conflict. Three things work in your favour. First, the city's best viewpoint, Mirador de San Nicolás, is free, open all night, and puts the Alhambra directly in front of you across the Darro valley. Second, Granada still has the free tapas tradition: every drink ordered at a bar comes with a small tapa at no extra charge, which means lunch and dinner can cost less than €10 a head if you drink at the right bars. Third, the historic centre is walkable, the monuments are clustered, and the Albaicín (a UNESCO World Heritage quarter) costs nothing to wander.
The main expense is the Alhambra itself. Tickets run €18 for adults (or €10–14 for partial access without the Nasrid Palaces) and must be booked weeks ahead in summer. That's unavoidable if you want inside. But the Alhambra view from San Nicolás, the 14th-century funduq at the Corral del Carbón, the silk market alleys of the Alcaicería, and the Albaicín quarter itself all cost nothing. A full day of sightseeing outside the Alhambra can easily be free.
For eating, skip the tourist-menu restaurants around Plaza Nueva and look for the tapas bars that locals actually use: Calle Navas, the streets around Plaza Trinidad, and the side alleys off Gran Vía. At these places a caña (small beer) costs €2–2.50 and comes with a tapa. Three drinks, three tapas, and you've eaten adequately for €7–8. Los Diamantes does legendary fried seafood at prices that haven't kept up with Granada's reputation. Chikito's menú del día runs around €12–14 for three courses with wine.
For sleeping, the two hostels in this guide (Oasis Backpackers and El Granado) both sit in the €15–30 range per person for dorm beds. Both are in the historic centre. Oasis has an Alhambra-view rooftop terrace. Neither requires you to compromise on location.