How can you disprove the statement $4=5$? I know this sounds insane. But bear with me, my friend said this to me with a straight face: 

Can you disprove the statement: $4=5$?

And I was like is that even a question, thats obvious, $5$ is $4+1$ and $5$ comes after $4$.
He was like: but you haven't still disproved my statement.
Something like this went for around 30 minutes...and he wasn't happy at the end at all
Is he correct? If not, how does one actually give a good sensible answer to such a person?

A good sensible answer would have to be based on some assumptions, i.e. on a set of axioms. The simpler the axioms, the better, but you cannot get away from some kind of assumptions or axioms. 
In truth, this situation holds for any theorem of mathematics, not just the theorem $4 \ne 5$. In calculus, for example, the assumptions one often uses are the axioms for the real numbers; from those, one can produce a proof of any theorems you run across in a calculus text.
In simple integer arithmetic, the assumptions one often uses are the Peano Axioms. It would be straightforward, albeit perhaps a little tedious, to produce a proof that $4 \ne 5$ using the Peano axioms.
One thing that happens with proofs is that in addition to having a long tedious formal proof, you also try to boil the proof down into basic intuitive statements. I'll venture to guess that in the end, the formal proof using Peano's Axioms that $4 \ne 5$ will probably boil down to exactly what you said, "$5$ comes after $4$". 
