A good low priced option for your first and last nights in Costa Rica. They have private rooms with bath, private rooms with shared baths, and dorms. Includes coffee and bread in morning in their enclosed backyard garden. There is a kitchen you can use and low-priced drinks can be purchased. Computer with internet access and wifi are available. There is a secure front open air patio with table and chairs and an interior living room with couches and large screen tv for all guests to enjoy.
It is a short walk to anything you need and Alajuela's central park. Restaurants across street and nearby with a wide price range and good food. Big used bookstore across the street. Police station is right around the corner. Hotel Cortez Azul is only a couple blocks from the central bus station where there is a steady stream of buses to and from San Jose.
Alajuela is a big, busy town that is lower cost than other areas with the problems that come from economic distress so you should also be careful there. It is not a place to just wander around at night but is a good place to stock up on things you anticipate needing/wanting like sun lotion, food, or alcohol.
Alajuela is also a good base site to visit the Poas volcano and coffee plantations.
A short drive from the airport, the taxi costs ~$6. Official airport taxi drivers know where it is and of course the Alajuela taxi drivers know how to get to the airport! They are driving by all the time, or ask Ramon and he will schedule a pick up at the time you need.
Ramon, the owner, and his staff are genuinely helpful and kind.
There are plenty of upscale, expensive options near the airport, it is more difficult to find lower cost, clean, and reliable places to stay. Hotel Cortez Azul fits that niche nicely.