I think you'll need 8GB RAM to play the new client. That's what the patch notes said at least.
(I'm having to buy more myself ...)

I'm sorry to say that while the ESO client will certainly profit from 8 GB system RAM or more going forward, 4 GB will, as stated in the patch notes, impact the game's performance negatively. The situation on the MB Air is even more unfavorable, as parts of the system RAM are used as video memory, further diminishing available space for the game.

You might want to try how the current PTS client behaves on Minimum or even Low quality settings. OS X 10.11 is the best choice, as it is a tad snappier than 10.10, as is the shader compilation when compared to 10.9 (the minimum requirement).

Any Mac that can run Yosemite is supported by El Capitan as well. Here's the full compatibility list. The upgrade should be offered (for free, ofc) in the App Store app.

Before upgrading, a backup via Time Machine or other means like Carbon Copy Cloner to an external HHD or NAS would be advisable, just in case. After backing up, the shell command (in Terminal.app)

sudo tmutil disablelocal

will clear the space of any remaining local TM backups. This might be helpful, as the PTS client takes up another 51 GB in addition to the current live one. ("sudo tmutil enablelocal" will re-enable local backups, which are made as long as Time Machine is generally enabled in System Preferences, and the MBA runs on battery).