Can I replace my laptop’s CPU?

Sorry the CPU chip is not really replacable as its soldered down. It really requires very special tools to be properly done.

You could replace the full logic board! MacBook Pro 13" Unibody Early 2011 Logic Board Replacement but you are limited to what Apple offered for this series as each series is logic board is unique!

Why do you think you lost one of the processors cores? More often a slow system is a sign of other issues as the system will go into CPU Safe mode to protect the logic board by ramping up the fan and lowering the CPU’s clocking.

Let’s check things out! Install this great app TG Pro you’ll want the full version to get the diagnostics. Sadly Apple killed it’s onboard diagnostics with the newer firmware. You could give it a try just the same, restart your system and press the D key. You’ll likely get an SMC error, Post a snapshot of the TG Pro’s main window so we can see all of the sensors and the diagnostic section here for us to see Adding images to an existing question

Reference: Mac startup key combinations