What is the best way to polish an old cocktail shaker?

Hagerty 10120 Silversmiths' Silver Polish, 12 Ounces

Silversmiths' Polish with R-22 tarnish preventative - protects as it polishes. This trusted formula cleans and protects your finest metals, including silver and gold, with its special R-22 tarnish preventing agent. It even cleans items with porous... read more...

This would be a good one to use because it also works on non-porous items, as I imagine a cocktail shaker would be.

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Reply · Send thanks · posted 153 days ago · 1 person agrees

Weiman Products LLC 014 Silver Cream

Make your silver shine. Wright's silver cream is the original, gentle, all purpose polish for all silver. It removes tarnish quickly, leaving a brilliant and even shine. It also works beautifully on stainless steel, chrome, porcelain, and much more. read more...

Not sure exactly what your cocktail shaker is made of, but I would think any good silver polish would do the trick.
This is a good polish that I use.

Reply · Send thanks · posted 153 days ago · 1 person agrees

Nevr Dull Metal Polish - - Bci

"Nevr-dull is an extraordinary cleaning wonder that shines all metals with sparkling ease.Each can contains a generous supply of specially treated cotton wadding cloth.Simply remove a piece of wadding from the can and rub it thoroughly over the su... read more...

This stuff is amazing. I use it on copper, silver, steel.. any metal that tarnishes. It has a fiberous product, soaked in its formula, that you tear off in small amounts and work your metal.

It does its job quickly, and looks great. (Plus it smells pretty good too :/ )

Reply · Send thanks · posted 98 days ago

Flitz SS 01306 Clear Stainless Steel Polish - 16 oz. Spray Bottle

This is a premium water-based stainless steel and chrome polish and provides a brilliant, protective "showroom" gloss. This can be used on stainless appliances, elevators, gas grills, vent hoods, and even works great on porcelain, enamel finishes,... read more...

This is a great product for polished different types of stainless steel, whether it's shiny or matte finished. It's totally water based, anti-smear, no oily left overs, has a decent smell, streak free, dust proof, and it won't leave fingerprints behind. It's pretty cool. All you do is spray it on and wipe it off. Works great.