just/even

[Taka]

What is the difference between these two? No difference at all?

When I just drink a cup of wine, I get drunk.
When I drink just a cup of wine, I get drunk.

Plus if those justs were replaced with 'even', how would the meaning of the sentence change?

[English Freak]

Here you want to emphasize on the fact that you get drunk with only a cup of wine, so put "just" before "a cup of wine". I think No. 2 is better.
And if you want to use "even" in your sentence, rewrite it this way:
Even a cup of wine makes me drunk.
or
Even with a cup of wine, I get drunk.
or
I get drunk even with a cup of wine.

[Taka]

So to you these sounds awkward?

When I even drink a cup of wine, I get drunk.
When I drink even a cup of wine, I get drunk.

[English Freak]

Not awkward, but prefer to say:
Even when I drink a cup of wine, I get drunk.
or
I get drunk even when I drink a cup of wine