Yes you can say "send this letter by air" although it's not as common as "air mail". Air mail is just a way of sending something similar to sending something "next day delivery". "Send this letter by sea" is not something you will hear very often as there are far more efficient ways of shipping things nowadays. Specially a letter.

Your alternative is grammatically correct and would be be understood perfectly. If I went to a post office in England, I would usually say "can I send this letter (by) airmail?" You can include or leave out the "by".