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If you took screenshot of your webpage with a monitor resolution of 1600x1200 then set the dpi at 300 by default it's going to print out at 5.3x4 inches. That's assuming your printing software respects the scale of the dpi and doesn't resample the image to fit into another physical size.
What I would suggest is finding the largest monitor you can and set the resolution as high as it will go. Then ttake your screenshot, you may even want to adjust the font sizes to a respectable size if they are too small. Drop it into a image editing application and change only the dpi to 300. You'll still be sending them the same image regardless of what the dpi is set at.....
One of my pet peeves is that a printing houses will require 300DPI but that number is really useless unless they also specify the printed size if you're working with lower resolution images. I ran into a similar problem where getting a size in inches they wanted was like pulling teeth. I have specific method for scaling images up that I use for screentshots of video which is much lower than a monitor. Done properly you can scale an image up in pixel size with some good results. If they provide the exact image size they need in inches you could scale it properly yourself with the right tools and techniques. Without the size they need its almost usleless as they will be resampling it anyway.
The 300 DPI is print resolution requirements so I'm not sure if I'll be able to get this to look good or not.
Let me explain again, if you take a screenshot of your monitor the image is going to be either 72dpi or 96dpi. For arguments sake lets say its 100dpi. If you go to print this image by default it will print at 16in.X12in. The width is 1600, divided by 100 you have 16 which is where the 16 inches comes from. The printer(your computer printer) is going to print 100 x100 pixels in a square inch.*
If you set the DPI to 300 its still a 1600x1200 image but now we are dividing by 300. By default it will now print at 5.3in. X 4in. The actual pixel size of the image has not changed, you've only changed the default scale that it prints at.
The other way to look at it is DPI is a translator, it acts as the middleman between the digital world which uses pixels as a measurement and a physical size in inches the image is printed at. Again, a requirement for a 300dpi image without a requirement for inches is really useless. Without that information they are going to have to resample your image anyway.
I'll give you another perspective going the other way, suppose you scan a 6X4 photo at 300DPI, the resultant image is going to be 1800X1200.
*note that these are defaults, printing software or applications can resample too and change the DPI or size of the image on its own...
As far as the quality goes make sure you are using a good image application, secondly use bi-cubic as the method for resizing. However you're going to be creating a larger fuzzy/softer image if you're scaling up.
They do have have plug-ins for photoshop and other applications specifically for scaling up, genuine fractals is one. These plug-ins examine the image for edges of contrasting color, it preserves them which will avoid the fuzziness.
I don't use them because I'm too cheap :P I scale in small steps using a program called neat image to clean out the softness. Works nearly as well but a lot more work.
If for example they require a 300dpi image at 3in X 2in. 800x600 would be more appropiate. If they haven't given you a specif dimension then I'd contact them and explain the problem. Scaling the text up is not much of problem but if you need to scale it down it will be, at some point the text will become illegible.
BTW just in case you are don't save as .JPG , save as .tif , .bmp or some other lossless format. .jpg is lossless and edges on text will take the hardest hit. Check with the printer they may want a different format.
If you really wanted you could recreate the pages in AI which you could send to them and it will scale accordingly. :) That's a lot of work.