Famagusta is best visited in spring (April–May) or autumn (September–October), when temperatures are ideal for walking the walls and exploring the old town (20–28°C). Summer (July–August) is hot — regularly above 35°C — and the coastal position means humidity as well. The walled town can be stifling at midday in July.
Winter is mild by northern European standards (10–15°C) and sees very few tourists. The main sights remain open. If you want the old city largely to yourself, November or March are good options.
Famagusta has no major festivals that most travellers would time a visit around. The main consideration is simply temperature and how much heat you can tolerate while walking on stone.