If a character who otherwise meets the requirements for an improved familiar doesn't possess the proper type or subtype, that might be okay. The player should
Ask the DM if the Character Must Have the Air Subtype First
Before trying to fit the air subtype on a character solely to have a mini-tornado following around the character, check with the DM. The feat Improved Familiar originally appears in the Dungeon Master's Guide's Characters chapter. In addition to providing information about those totally optional prestige classes, it includes such things a primer on the epic rules and playing monstrous races. The whole chapter, then, is about making the game do what the DM wants it to do with regards to the PC. That puts the ability of any character to even take the feat Improved Familiar (DMG 200), and, subsequently, what creatures are available as improved familiars, more than a lot of the game's other optional-but-considered-available rules, completely in the DM's hands.
That said, the text you quoted provides a way out. Again, it says
The list... presents only a few possible improved familiars. Almost any creature of the same general size and power as those on the list makes a suitable familiar. Nor is the master’s alignment the only possible categorization. For instance, improved familiars could be assigned by the master’s creature type or subtype, as shown in Table: Improved Familiar by Type/Subtype.
Emphasis mine. So, y'know, the improved familiar totally could be assigned by the master's creature type or subtype, but the improved familiar doesn't have to be, and the DM can just say they're not and use the table as-is except ignoring that column. That would allow any character eligible for the feat Improved Familiar who qualified for a familiar with an alignment of N and who was a 5th level arcane spellcaster to take an air elemental as an improved familiar after dismissing his current familiar (if any). It's unlikely this would break anything, and other, later improved familiars largely ignored the possibility that the DM could, instead, assign possible improved familiars based on the arcane spellcaster's type or subtype anyway.
Gaining the Air Subtype after Character Creation Is Challenging
Barring taking a class that turns the character into an elemental (and several do but are often lackluster--e.g. elemental archon (FP 190), elemental savant (CAr 32)), and short of just becoming a different creature possessing the air subtype via shenanigans like magic jar [necro] (PH 250-1) or polymorph any object [trans] (PH 263), the easiest way to gain the air subtype appears to be the Ritual of the Elements on page 148 of Savage Species, which is still a bad deal.
Other than that, it's homebrew stuff, like a newly discovered Afelbain gem (Dungeon #112 91) that grants the air subtype like the two extant minor artifacts grant the cold and fire subtypes or a spell like the 8th-level Drd spell mantle of the fiery spirit [trans] (Sa 118) but granting the air subtype instead of the fire subtype.