I would like to know the name of the airport in the city of Lagos and the closest hotels with the rate below $250
Closest airport would be Faro Airport. Depending on your departure location, the majority of the flights will be landing in Lisbon though and then you will probably have to buy an extra flight to Faro. I would recommend you to look at the bus and train options as it might be more cost friendly. A bus trip to Lagos should take around 4 to 5h.
For the hotels, worry not. There are plenty of great options under $250. Check expedia, booking or airbnb for more info. You will find great hotel options around $100.