How to upgrade hardware on my device

While this sounds like a lot that is needed breaking it down to a couple of repair sessions can make this less daunting. Your system still has a long life ahead of it.

Your battery is likely worn out given your systems age. Here’s the needed part MacBook Pro 13" Unibody (Mid 2009-Mid 2012) Battery and here’s the needed steps Sostituzione batteria MacBook Pro Unibody da 13", versione metà 2009. Thats likely the reason your system shuts down even if the system is under AC power.

Your systems drive is likely in bad shape. There is two directions you can go here. The easiest is just replace your drive or try to recover your hard drive

Replacing your drive will require finding a drive which is compatible with your system as your systems SATA port I/O speed is only SATA II (3.0 Gb/s) I would jump to a SSD which will make your system sing! Here’s the best drive here Samsung 860 EVO looking at the spec sheet here we can see the Interface the drive supported is SATA II (3.0 Gb/s) many SSD’s today are fixed speed SATA III (6.0 Gb/s) which won’t work reliably here. Here’s the needed guide Installazione dell'hard drive nel MacBook Pro Unibody da 13", versione metà 2009 I would take the time to replace the HD SATA cable as the older cables tend to have issues Your Hard Drive Cable Is A Ticking Time Bomb here’s the needed cable MacBook Pro 13" Unibody (Mid 2009-Mid 2010) Hard Drive Cable At least place the electricians tape on the uppercase to help protect it and don’t handle the cable to much as they get brittle with age.

You’ll want to re-install your OS and restore your apps and data from your old drive using this StarTech 2.5" SATA to USB adapter cable so you can setup your new drive (jump to the OS install as the end)

The last issue is dust and thermal paste, again because of your systems age your system is likely hitting thermal issues which is why under stress the system shuts down this is most notable when you hear the fans racing.

Get a small soft paint-brush to carefully scrub the surfaces to loosen the dust (it doesn’t take much pressure) using a can of can’ed air to blow the dust and other junk away. Follow this guide to remove the logic board MacBook Pro 13" Unibody Mid 2009 Logic Board Replacement and here’s the guide to clean the old paste off Come applicare la Pasta Termica. You don’t need fancy thermal paste [prodotto collegato mancante o disabilitato: IF179-010] is just fine here. Also don’t over do the paste!

All you need is enough to fully cover the chip die its self.

The last issue is re-installing the OS. Here we do have a slight problem as the OS-X El Capitan (10.11.x) is the highest this system can support. Before you take out your drive or remove the logic board you’ll want to alter the systems data back dating it to the OS’s certificate window as the certificate won’t allow you to install the OS with todays year! Set your systems year to 2016 that way your OS installer still works.

Here’s a bit more If you've got an old macOS install image, it will probably stop working today which deals with the newer versions of macOS which Apple forgot to update Sierra, High Sierra & Mojave certificates as they mis-set the expire date.