In the LISTS table, there is a list name called 'Requirement Type' but that is not what the Type dropdown uses. There is another table called REQ_TYPE that has the types on the dropdown list. These req types have ID numbers, and this ID number is how they are ordered on the dropdown. The REQ table stores this ID for each requirement in the RQ_TYPE_ID field, and as far as I can tell, the built-in Req types and their ID numbers are fixed and should not be changed. (I did try changing them to re-order the list and it caused problems, probably since this ID is referenced in other tables.)
When you add new req types via Project Customize and the Requirement Types screen, the IDs are assigned sequentially. I tried renaming the default types (you are allowed to rename some of the types but not all) to make them alphabetic but that did not change their order on the dropdown. Again the order is determined by the ID number, not the name.
So the only way to re-order the dropdown list that I can see, is to create new req types in alphabetic order and Create them As each of the built-in types. The names of the new types have to be unique, so you will have to be clever if you still want a requirement called 'Functional' (a built-in type) but want it to appear alphabetically in the dropdown.
IMO this is a lot of work for just the benefit of seeing an alphabetic Req Type dropdown list. My observations apply to v11.00 but I suspect it applies to other versions as well.