You won't achieve the same look because cherry is a closed-grain wood and oak is open-grain so the grain will look very different. However, you can work to get a similar color. Staining will highlight more of the grain and let some of the natural grain color come through, where paint will cover more of the grain (though because oak is open-grain, you will still see some of it).
Rustoleum makes a product that might help tranform the old oak cabinets (if you're doing it yourself): http://enzyliving.blogspot.com/2011/02/rust-oleum-transformations-another-diy.html
Also, see how we transformed my parents' kitchen by painting the old oak cabinets. http://enzyliving.blogspot.com/2010/02/mom-dads-kitchen-facelift.html
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