This has little to do with the preposition at and more to do with the way the nouns are being used.
A noun like school can be used two different ways:
- As a concrete noun referring to a physical building (or complex of buildings) in the real world. "The father went to the school to meet his daughter's teacher."
- As an abstract noun referring to the general concept or activity of primary learning. "Most American children go to school for twelve years."
Abstract nouns use no article, whether they are in a prepositional phrase or not:
- School was difficult for him.
- She spends most of her day at school.
- Hurry up or you'll be late for school.
Concrete nouns do use an article, whether they are in a prepositional phrase or not:
- A school is being built downtown.
- The school is on fire!
- I will meet you later behind the school.
Proper names, as you note, also do not use articles:
- Microsoft released three new products today.
- I got a job at Microsoft.
Note that home is always an abstract noun, except in the meaning of a convalescent or retirement home. A house or an apartment is a concrete place, but home is the abstract concept of place being a primary residence.