Do better than The Sun or upstage The Star? (8)

I believe the answer is:

outshine

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'do better' is the definition.
'outshine' can be an answer for 'better' (thesaurus). I am unsure of the 'do' bit.

I can't explain the rest of the clue.

Can you help me to learn more?

(Other definitions for outshine that I've seen before include "Do better than, being more brilliant" , "beat" , "Eclipse" , "Be much better than" , "Surpass in splendour like the sun" .)

I've seen this clue in the Irish Times.

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