You cannot "change" from Photoshop to Illustrator. You will still need both of them. For example to achieve this: rotate and crop the image in Photoshop.
You can also create a clipping mask in Illustrator, but that won't produce a smaller file size.
If you want to do it anyway: draw a rectangle the size of the artboard, make sure it's on top of the image, select both and Object > Clipping Mask > Make
As far as I know, any cropping (or other similar change) of a JPEG will resulting in a resampling and thereby a certain deterioration (you can see it clearly in old JPEGs that have been round for a long time).
If you want to do it in Illy, you may use the dirty destructive deed (which at least one member rightly describes as hideous):
1) Create the mask as described in the first post;
2) In the Transparency panel dropdown select something else than Normal;
3) Object>Flatten Transparency.
Edit: I was away for a while before pressing the Post Message button. Hi NN.