To what extent is the university budget public? I guess this is technically a legal question, but since it is about universities I hope it is ok to ask it here.

Let's say that university A has employed the construction company X for a big project. Let's also say that from an outsider's perspective the deal seems a bit dodgy. How much of this information has to be public? In other words, what can a general member of public learn about this deal?

Assume that university A is a standard "public" UK university and company X is just a big private company.
The key to this information request would be the Freedom of Information Act. The act has lots of things which you cannot request information for, however. One of them is "trade secrets". I don't know if construction work can be considered a trade secret.

Timothy Gowers documented his attempt to get UK universities to reveal their journal subscription costs on his blog. This sounds relatively similar to what you're thinking of. If you attempt to get the same information, I'd guess you'll need to use the same methods, will meet the same obstacles, and get the same results (i.e. you'll eventually be successful).