Regardless of the time of day you decide to fly, the ticket price will remain around the same. Book flights from San Jose to Nairobi as you normally would.
Due to multiple factors, expect to potentially pay more when booking in June. For low rates, May can be a great choice for travel to Nairobi. Flights to Nairobi from San Jose in May are about $936 on average during May, but can be found for as low as $621.