The resolution of your laptop's display is irrelevant unless you want the image cloned on both screens at the same time (which I doubt is the case?).
The reason why your image doesn't look very sharp is because you have not got it set at the native resolution. Do this and you will see a big improvement. If you find things too small then try adjusting the DPI within your display settings, and on websites use your browser's zoom feature (press ctrl & + on most browsers) - the result will be much better than using 1024 x 768 as your resolution. If you use Firefox, it will remember which zoom settings you use for each site, so you will only have to do it once. For example, on this site I like it quite zoomed in as it's very small on my monitor otherwise, so when I come here from say Google, it will go from zero zoom to my custom zoom automatically. Internet Explorer on the other hand will just stay at the current zoom level for your entire session unless you manually change it.
Buying a new monitor isn't going to help - all monitors produce a poor image if not used at their native resolution, and I suspect that a 1280 x 1024 19" monitor will have among the highest pixel sizes available.