Over budget or Under budget are not idiomatic expressions. Idiomatic expressions are informal English expressions, examples of this, using the verb budget would be "Champagne taste on a beer budget" meaning that you have expensive tastes that are beyond your financial means or "Shoestring budget" meaning you have a very small budget to work with. Over and Under are prefixes that can be attached to a verb like budget making the verb an adverb. To answer your question, yes you can say "Over budget" or "Under budget". For example "He went over budget this last month" or "He plans to come in under budget for the new car". I hope this help.

Over budget, under budget, on budget are all terms, maybe compound words. You can say "over the budget" but it is more common to say "over budget". Over budget is more like an adjective, you would more likely say "over the budgeted amount" if you use "the". There is no difference to the meaning. If you are talking about something not being affordable for you personally - you can say "it's out of my budget" or "over my budget". There is an idiomatic expression - "champagne taste on a beer budget" - meaning you want/like things you can't afford to buy. :)