MacBook Air as primary monitor for Windows PC?
Only certain (mostly older) iMacs could be used as an external monitor for another computer; see the answer at this question as a starting point: What are the requirements to use iMac screen as a second monitor - Target Display Mode?

Your best bet would probably be to:

Have the PC and the MacBook Air on the same wired or wireless network,
connect to the PC with Microsoft’s free Remote Desktop client (Instructions / More info &amp; download link), using the IP address of the PC and your Windows username/password.

Notes:

You will probably need to use a HDMI television, borrow a monitor from someone, or move the hard drive/SSD to a PC with a monitor in order to do the initial setup on the Windows side.

Due to security issues, this will no longer (currently) support the advanced features (e.g. 3D acceleration) of your PCs graphics card, but as a ‘budget’ PC that will likely not be much of an issue.

You can even use your MacBook’s built-in camera and speakers with the PC, and if you connect a second monitor to the MacBook then you can use both screens with the PC. It just has to happen over the network.

If you are using Linux instead of Windows on the PC, you’ll need VNC instead of Remote Desktop, and you won’t be able to share as much of the MacBook’s hardware with the PC.