550 battery cycles isn't super high...usually Apple says portable computer batteries should still retain 80% of their charge capacity at 1000 cycles. But the "Service Battery" message is concerning.
Yes the battery can be replaced. Here's one example of a replacement battery for a 2013 MacBook Air (assuming it's a 13" display model):
NewerTech NuPower 60 Watt-Hour Replacement Battery for all 13" Apple MacBook Air 2010 - 2017 models. 1 Year Newer Technology Limited Warranty.
eshop.macsales.com
Before spending any money...I would suggest doing an SMC Reset:
If your battery won’t charge, check the software and hardware of your Mac, or get your Mac serviced.
support.apple.com
HTH,
Nick