I try to prove that every ordinal number $(\alpha,\in)$ is well-ordered, where an ordinal number is defined as a transitive $\in$-linearly ordered set.
So all I have to show is, that every non-empty subset of $\alpha$ has an $\in$-minimal element. That follows quite simple by axiom of foundation, but how can I prove this without foundation?
There's a good reason you're having trouble - this is not provable without Foundation: it is consistent with $ZFC^-$ (that is, $ZFC$ without Foundation) that there is a sequence of sets $a_i$ ($i\in\omega$) such that $a_i=\{a_j: j>i\}$. Then $a_0$ is transitive and linearly ordered by "$\in$," but is clearly not well-ordered by "$\in$".
The existence of such a sequence is a consequence of Boffa's Antifoundation Axiom. This axiom is less popular than Aczel's Antifoundation Axiom (AFA), and it seems harder to get a good exposition of it. If you can read French, Boffa's original paper is http://www.ams.org/mathscinet-getitem?mr=476514; otherwise section 1 of the paper Standard foundations for nonstandard analysis has a good exposition.
I suspect that AFA also provides counterexamples to your claim. However, this is a bit trickier; in fact, AFA disproves the existence of the counterexample I describe above! (In AFA we have $a_0=a_1=...$, so the result is a singleton - a (in fact, the) Quine atom.)