That is correct, I think of 1245 and 1250 as subsets of 1231 property. Sec 1231(b)(1) includes property "of a character which is subject to the allowance for depreciation provided in section 167, held for more than 1 year".
Section 1250 applies since 1250(c) states "the term “section 1250 property” means any real property (other than section 1245 property, as defined in section 1245(a)(3)) which is or has been property of a character subject to the allowance for depreciation provided in section 167."
Therefore, the undepreciated portion of the building and land are taxed as capital gains.
Because of the above definitions, the rental does not need to rise to the level of a trade or business to be classified as either 1231 or 1250 property.