Question: We know that a cat is able to endure, continue, or survive despite a near encounter with death or disaster because cats have nine lives (according to a common myth). 

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> Example: 
>  - Mr. Pickles has been missing for a few days, but I wouldn't worry about him. **He is a cat with nine lives.**   
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Does my bold self-made sentence work properly and idiomatically here?

Answer: 
It is in the past tense. This is a "[future in the past](https://www.englishpage.com/verbpage/futureinpast.html)" construction. The question is rhetorical. Leonard isn't looking for an answer, but he phrases his statement as a question. Changing it back to a statement gives:

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> I told you I would be working nights.
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So in the past Leonard said "I will be working nights". In the past Leonard used a future tense. When he reports this the tense of "will" is changed to the past tense. The past tense of "will" is "would".

This construction is quite rare, it is most likely to appear in reported speech, thoughts or emotions 

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> I knew it would be fun.
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Leonard could have used a "going to" for the future with little or no change in meaning.

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> Didn't I tell you I was going to be working nights?
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It's not "would be doing", it's "would be sth".

`Working nights` is a idiom means the shift of working at night.

"Would" indicates that all the "L telling S about working nights thing" happened in the past, not this morning, it might be yesterday.