Actually you can remove the drain from the bottom of the tub. You need to purchase a special tool for its removal from the hardware store. Its a plug wrench. It has a t-cross that fits over the cross bar inside the drain. The wrench is hex shaped with two different ends for larger and smaller size. If you use this tool, you can remove the drain and take it to a hardware store for replacement. I would put a lift and turn in its place. Generally the threads on the chrome drain are standard (though a few are different). There are other ways to remove the drain, but you will find this is the easiest.
Once you get the new drain, use silicone, instead of puddy, to seal it and you will not have any leaks.
As far as the sink, purchase a new 1-1/4 pop up kit from the hardware store when you go. The old one can be removed with a set of channel lock pliers (water pump pliers). Again follow the directions in the package for its installation. And do not buy a 100% plastic pop up kit. You will not be happy in the long run.
Use silicone again instead of puddy for the seal on the sink. Most marble molded sinks recommend not using puddy, because it causes the sink to crack. Puddy and the sink are both petroleum products and the puddy causes the sink to break down.