According to this question, the pullback in Cat exists by ordinary abstract nonsense. I've only built pullbacks explicitly in specific contexts such us the fibre product of schemes, pullback of sets and topological spaces... but these constructions required working on the specific category.
Where can I find the proof that they exist in Cat? For my purposes I only need it to exist in the subcategory of additive categories, but if it exists for all small categories then it's even better.
Cat is a bicomplete category i.e the category in which all small limits and colimits exist. (For reference regarding this fact, please check the examples of bicomplete categories in https://en.wikipedia.org/wiki/Complete_category) A pullback can be represented as a particular limit of a diagram. Hence 1-categorical Pullback exists in Cat.
Whereas when Cat is considered as a strict 2 category, Construction of the corresponding 2-categorical pullback in Cat is mentioned in https://stacks.math.columbia.edu/tag/003R