The answer is correct except for one detail and that is that the person answering would not repeat the word “travail” in most cases, although it is not incorrect and can be done at times (especially with certain connotations), and would instead use a pronoun in its place.

• Non, je ne le finis pas aujourd’hui.

Nevertheless, this basic answer is not the most usual one. Often enough, someone answering such a question will use the future tense or the form “aller finir”, which is used to express actions in a near future (futur proche) or also future anterior.

• Non, je ne le finirai pas aujourd’hui.
• Non, je ne vais pas le finir aujourd’hui.
• Non, je ne l’aurai pas fini aujourd’hui.

There are other possible variants.

• Non, ce n’est pas aujourd’hui que je le finis.
• Non, ce n’est pas aujourd’hui que je le finirai.
• Non, pas aujourd’hui, je ne le finirai pas.

There is as well a very short variant, which follows ;

• Non, pas aujourd’hui.

Of course a simple “Non !” is also an option.

Correct.
I would say either:

• ‘Non.’ to simply answer the question
• the sentence as you’ve mentioned (‘Non, je ne finis pas mon travail aujourd’hui.’) if I want to insist on the words used in the question.