Quick answer: The best time to visit Barcelona is May–June or September (warm, beach + fewer crowds). Expect a mid-range budget of about €95 per day. Visa: Schengen area — 90/180 rule applies for non-EU passports.
Why this window
Barcelona is walkable if you cluster Gaudí, the Gothic Quarter and the beach rather than crossing town. Four days covers the icons plus a calmer Montjuïc afternoon.