Before and after a Shadow Link battle, your health will be restored. Also, if you lose, it doesn't count as a Game Over, so you've got nothing to lose by fighting, except maybe some time.
The battle will take place in a variety of arenas, depending on the Shadow Link's location. For instance, if he's at Turtle Rock, you'll fight in a fiery arena. Shadow Link fights just like a normal Link would, using the exact same moves. Unlike other monsters, he's very aggressive and knows how to defend and manoeuvre around attacks. At the beginning, we'd advise against fighting Shadow Links with a higher bounty that yours, because you may lose. However, once you've had a few battles, you should become familiar with Shadow Link's behaviour.
To win, simply reduce the Shadow Link's heart count to zero. Our general tips for fighting are to keep moving and don't stay in the same place for long. To make life easier, bring an item that can stun, such as the Hookshot, Boomerang or Tornado Rod. Even though Shadow Link is smarter than your typical monster, at the end of the day he's still an AI and can be exploited. Wait for Shadow Link to attack, then look for an opening and immediately stun him, then hit him when he's vulnerable. Repeat for an easy win.
Either way, after you emerge victorious, you will earn the Shadow Link's bounty and you may also unlock achievements, known as "Challenges".