Here's a link that my help -

Removing Rust Stains from a Sink http://www.thriftyfun.com/tf338534.tip.html

That would be unfortunate if you have to replace that beautiful tile. Old-fashioned is very cool - having a vintage element in a contemporary home works beautifully. I would switch out the hardware and have the rust fixed and the bowl painted.

You might be able to get some help from the Retro Renovation site:

http://retrorenovation.com

All good questions for a plumbing shop? I think. You could try Lowes or Home Depot. Maybe it's more of a tile question. How to keep the tile intact may be the more complicated part of this.

That is a tile in sink. To remove it, you will need to remove the tile surrounding it's lip. Not going to be easy to do and keep the tile intact to reuse. You need someone with some skill with a multi tool. And a backup plan. As in what to do if you damage the tile and cannot find replacements in that color. Daltile does make some of the trim, but not in that color, and it is a long wait special, order.

Have you tried Bar Keepers Friend on the rust? I would clean up the rust and vow to never fill the sink. That is the only way water will get in the secondary drain. It is a shame to spend money now when you think you will remodel in under three years. Is water getting under the sink?

Since you want to change everything in a few years, I suggest you try a cosmetic fix for right now. Use some Barkeepers Friend to remove as much rust as possible, gently. Then use a porcelain repair paint, available in nail polish sized bottles for about $5 from hardware stores, to touch up the rusted area. It will take two or three coats of somewhat fussy work. It will look much better even though it is not perfect.

For the faucets, assuming they work just fine, just buy two new chrome lever style handles for each faucet -- they look like they are standard faucets where the handles are easily removed and replaced.