The title says it.
When I try to export a png I get the following dialog, where I can choose the PPI, but not the resolution.
Playing with the PPI around to get the right size could probably work, but it's annoying. There must be a better way, since resolution independence is the entire point of vector graphics.
Best Answer
I have an older version of AI running on Windows, but with it the way I do what you want is by selecting Save for Web... on the File menu which opens a dialog of the same name. In that over on the right you can select the PNG output format desired, and then on the Image Size tab you can set the exact horizontal and vertical resolution of the image created.
This can still be a little tricky because when either using Export... or Save for Web..., the overall image dimensions/aspect ratio are determined by an imaginary bounding-box that encloses all the objects in the drawing. Because of this, it is sometimes necessary to add an empty rectangle around the (larger) area desired to force the proportions to be what is desired. Alternatively you can sometimes do this afterwards in Photoshop (or other image-editor).
Here's an illustration of using Save for Web... to create a 24-bit PNG file showing where to set the pixel resolution in the dialog: