You can reach Ekkerøy by taking direct flights with SAS or Norwegian Oslo-Kirkenes (approximately 2 hours flight). Then you can take Widerøe aircraft (approx 10 min flight time) over The Varangerfjord to Vadsø Airport, and rent a car there. It is about 10 min drive from Vadsø Airport to Ekkerøy.
The alternative is to rent a car at Kirkenes Airport and drive the magnificent trip around the bay. The tour takes about 2.5 hours and is 185 km long. It is easier to get hired car in Kirkenes than in Vadsø.
It is also an opportunity to travel by plane from Oslo-Tromsø and then take direct plane from Tromsø Vadsø.
Some also travel from Helsinki to Ivalo in the north, and rent a car there before they drive about 4 hours to Ekkerøy.
The ferry Hurtigruten have daily morning arrivals at Vadsø (which is 15 km west of Ekkerøy) and it is an alternative for those who have plenty of time.