Is AppleSupport@insideapple.apple.com a genuine email?

The message looks a bit odd, particularly the URLs referred to. I would recommend not following them. There is an Apple page on this subject:

https://www.apple.com/support/13-inch-macbook-pro-solid-state-drive-service-program/

so it looks as if the program is genuine (and it's been reported on by other websites), but the email itself may or may not be. The page I link to above gives you information on how to check whether your Mac is affected and how to proceed.

The only way to find out if it is real or a spammer. I would call the Apple store in your city and ask whom ever answer the phone and then ask the question that you have posted on this site.

Performing a DNS lookup on those URLs pointed to an IP address on Apple's servers, so they're real.

(169531)

Thanks Jiro I just spoke with Apple Care and this is definitely not genuine!

Are you sure about that? One problem with scam e-mails is that legitimate companies send out e-mail just like them. Scammers would be unlikely to have your machine’s serial number.

What you should do in these cases is go to the website yourself and verify this information. Don’t use any link in the e-mail.

In this case, this is the link you should check: https://www.apple.com/support/13-inch-macbook-pro-solid-state-drive-service-program/

Enter your serial number and, if it says you might have a problem, contact Apple support and schedule an appointment.