The single most important factor in timing a Patras visit is the carnival, which runs from mid-January to Clean Monday (the start of Orthodox Lent, typically in late February or early March depending on the year). The final weekend, with the Grand Parade on Saturday and the burning of the Carnival King effigy on Clean Monday, is the climax of the whole celebration and draws enormous crowds. If you want to experience it, book accommodation three to four months in advance for the peak weekend.
Outside carnival season, April through June and September through October offer the best combination of good weather, manageable crowds, and reasonable prices. July and August are hot — Patras sits in a relatively sheltered position that can feel airless in a heatwave — and the city is busy with travellers catching ferries to Italy. The Patras International Festival runs from July through August with performances at the Roman odeon, which is a reason to visit despite the heat. Winter is quiet and mild by northern European standards; the city functions normally, and the castle and museums are less crowded.