I have the 2009 white MacBook and am still very fond of it. It's running Snow Leopard, which I love. I also have both an eleven-inch and thirteen-inch MacBook Air. If like me you're still running an old OS on your trusty MacBook you may find the latest OS, Sierra, a little unfamiliar in some ways for a while. Other than that, the Air is a truly excellent machine and I imagine you'll be delighted with it.
The eleven- and thirteen-inch models both have their merits. While the smaller is fantastically portable and a little workhorse, I found that with use the screen size wasn't ideal for me though I wouldn't want to part with it. I tend to wear reading glasses now when I use it. The way the screen fonts and user interface are designed these days doesn't help matters. I don't like the 'modern' look very much and it doesn't seem to enhance the Mac's famous usability much but rather detract a little, in my experience, which I think is a shame.
The thirteen-inch is just right for screen size (and what I am used to generally), and I got such a great deal on it that I couldn't say no. I'm glad I bought it, and while I am beginning to accept that my white MacBook is coming to the end of its stalwart productivity for me, it's given me seven good years (and not finished yet). Indeed I've found that Macs have always provided great reliability and length of service.
I haven't found the quality of the Air's display (rather than the size) to be problematic, as some others seem to have experienced. Yes, you can get other machines with retina screens and that's great if it's what you want or need. The 'old-fashioned' displays still look fine to me, however, for what I need and want and overall I think the Air is a very good package. Incidentally, a friend used my thirteen-inch Air recently and thought the display was excellent and much superior to his Windows laptop's. He was also completely knocked out by Apple's design and build quality. Sometimes I think as Mac users we've been spoilt and now can easily overlook some of the strengths that the company's products continue to offer us. There are plenty of people still coming new to Apple's computers and discovering the advantages of being a Mac user. Yes, the Mac Mini and Pro are overdue some love from Apple, but as an eco system the Apple experience has a lot going for it.
Good luck with your upgrade.