type of tool?

[T.allan]

Starting of with the "Bottle Stopper". Is a 3/8" spindle gouge adequate for this small job?

 

[CHJ]

Short answer is Yes,
Long answer? how many hours have you got?

There are several dozen ways you can make such a simple item and all with differing tools and techniques.

Perhaps a thin parting tool may be easier for you to turn the shaft for the cork.

 

[T.allan]

Is it ok to turn the shaft or is a dowel adequate also?

 

[CHJ]

Why bother with drilling holes and fitting a dowel to length when 10 seconds on the lathe will do the same or better?

 

[T.allan]

OK thats great. It sounds like its better to do the 10 secs rather than dowelling which obviously takes time. Thanks for the contributions