Sure, I think you can just copy your index.php (maybe pape.php) but try index.php first. Once you have a copy of index.php change the name to something like secondstyle.php.
Then open the secondstyle.php and at the very top add the following code.
Just save the file and go into the admin area of wordpress. Then go to your ‘pages’ section and the click on the page you want to have a different style sheet. Then when your in the edit area of this page on the left bottom of the page you will be able to choose the new template file called ‘Second Style Sheet’.
The just save, and now that page when loaded should have that stylesheet you specified.
Also if that doesnt work you may need to copy a different page instead of index.php, and maybe do page.php. Someone else may know better.
Let me know if it works for you or if you have any other questions.