The French sentence might have a comma before mais and a full stop at the end:

Oui, j’ai pleuré, mais ce jour-là.

In such case, your first attempt is fine. The reference is not specially “today” though :

Yes, I cried but it was that day (not another day)

I guess you can avoid the second verb in English too:

Yes, I cried, but on that (specific) day.

However, the actual song (Laisse tomber les filles, by Serge Gainsbourg and sung by France Gall in 1964) words are:

Oui j’ai pleuré mais ce jour-là, non je ne pleurerai pas.

which means,

Yes I cried, but on that (future) day, no, I won’t cry.

I think the sentence is not complete. I suggest to you this translation:

Yes I cried, but that day ….

The complete sentence could be like this:

Yes I cried, but that day, I didn’t

Oui, j’ai pleuré mais ce jour-là

can simply be translated by:

I cried that day.