1) She did not come to work on my last working day.
2) She did not come to work my last working day.
Neither sentence is actually very good.
1 - is better because it's meaning is more clear. It is just two prepositional phrases, one after the other, adding information to the basic sentence "She did not come to work..."
2 - is incorrect because it's meaning is ambiguous. Is 'to work' a prepositional phrase or an infinitive verb? Without the preposition ON, the phrase 'my last working day' becomes the object of the infinitive verb.The infinitive 'to work' needs an object to complete its meaning. This become more clear if you substitute a different verb - 'She did not come to celebrate my last working day.' But even this sentence could misinterpreted. "She did not come to work on - could still be interpreted as leading to an object for the infinitive - ...on my project, she came to work on Claire's project. So, "my last working day" might be interpreted as something that could be worked on (but maybe that is a stretch!). A better phrasing for this, instead of 'my last working day' would be 'my last day at work'. But in any case you still need to preposition ON.