I'm going to elaborate on the answers given so far (the answer + associated comments above) in case you or someone needs more details.
Screen Sharing is what Apple calls the VNC / remote desktop technology built into OS X for many years now. I'm assuming your machine is running 10.10 Yosemite. Instructions for earlier versions of OS X are similar.
Screen Sharing allows the Mac--in this case the Mac pro sans monitor--to display on another machine on the local network; for you, your Air. Screen Sharing is configured in the Sharing pane of System Preferences. System Preferences can be accessed from the Apple menu at the top left of the screen, for example.
You may need to borrow a monitor or get help at an Apple store near you. I'd be surprised if they didn't let you borrow one of their monitors to configure your Mac Pro for Screen Sharing.
Here is what you need to do from your Mac Pro:
- Run System Preferences.
- Click on "Sharing" in the third row.
- Click the check box next to "Screen Sharing".
- Notice the green dot and "Screen Sharing: On" to the right of that check box. Underneath that note, it says: "Other users can access your computer's screen at vnc://x.x.x.x/ [though the x's will be replaced by numbers] or by looking for "[Your Mac's Name]" in the Finder sidebar."
- Take note of your Mac Pro's Name. Note that you can change your Mac's name at the top of "Sharing" in System Preferences if you want to.
- Make sure that both your Mac Pro and your MacBook Air are connected to your home network.
- Move to your Air.
- Go to the Finder (by clicking the smiling Finder guy on the Dock or by clicking on the Desktop background.
- Press the COMMAND key + n to open a Finder window.
- Look in the sidebar, the leftmost part of the window that shows a list of folders, disks, etc. Locate "Shared", a header that will only show up if a machine on your network is sharing files (and / nor Screen Sharing).
- Under Shared, you should see the name of your Mac Pro. Click it. You should see a button near the upper-right corner of your Finder Window.
- Click that button. That should launch the Screen Sharing application that will allow you to interact with your Mac Pro. Type in the username and password that you set up as your user account on your Mac Pro. Note that screen refresh will be slower than a real monitor and are dependent on the speed of your network.
Good luck! I hope this helps. I use VNC / Screen Sharing between my Mac and my iPad all the time. My Mac is usually in the basement, so if I need to look at something quick, it's often easiest just to use VNC (I use Remoter VNC) on the iPad. (For your case, might I humbly suggest that you consider buying a used monitor temporarily. You could probably get one for $50 or less and it would most likely be a better experience.)
- Note: I put in a lot of detail. I am not trying to insult your intelligence. I just didn't want to skip over any steps that might cause confusion.