The technical term for the part you're describing is a "pusher." Depending on how the pusher was installed--some simply screw onto a little internal post, while others are installed from inside the watch case, it may be an extremely simple or only pretty simple repair for a watchmaker. Parts supply houses sell boxes of assorted pushers that repair technicians can use for exactly this type of repair, and any place that does watch repairs more complex than simple battery changes is likely to have replacement parts in stock.
I would expect a replacement pusher to cost a few dollars (~$5 or less), plus the labor to install it. That, unfortunately, will vary tremendously depending on the number of people in your area who can perform this type of repair and the complexity of the repair itself. It should cost very little to simply screw on a new pusher from outside the case--that will take only a minute or two to complete. It will take more time to remove a movement from the case to service a pusher that's installed from inside the case. I wouldn't expect labor for that to be less than $20; some places might try to charge more.
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