I currently have a 23" Panasonic HDTV (specs listed below), currently
connected to an XBOX for DVD playing.
1. Is there any way to improve the picture with an upconverting DVD
player given I have neither a DVI, nor an HDMI connection.
2. In general, should I buy a stand alone budget DVD player (~$80)?
Will that be better than the XBOX?

Composite AV In (RCA): 2 
Composite AV Out (RCA): 1 
S-Video In: 2 
Component Video In: 1 
RF In: 1 
Headphone: 1

You really aren't going to see the benefit of an upconverting DVD on a
23" TV. An HD screen that small should have a great picture
regardless.  You would have to be quite the videophile to be able to
notice any difference.