Lugo is a year-round city — the old town functions regardless of season. The best time to visit for weather is May to September, when temperatures are mild and evenings warm enough to sit outside. July brings the Arde Lucus festival in the last two weeks of the month, which transforms the old town into a Roman and Celtic reenactment event drawing hundreds of thousands of visitors — book accommodation well in advance if you're visiting then, or far in advance if you want to come during the festival itself. The San Froilán festival in early October is the city's own main local celebration.
Galicia is one of the wetter corners of Spain, and rain at any time of year is possible. The combination of stone arcades and the enclosed Roman walls means the old town remains walkable even in damp weather.