What does "shine" mean in the context of this Harry Champion song?
Solution 1:
The figurative use of the noun shine OED going back to ~ 1560
brilliance, radiance, splendour, lustre.
As in:
1867 J. R. Lowell Fitz Adam's Story No other face had such a wholesome shine.
And from Ahdictionary.com
excellence in quality or appearance
So, to shine up your appearance and/or illuminate it with splendor.