Well, there are a lot of questions to answer. If it's the same as this one -
http://classicfuji.com/Tivoli_1988_Page.htm - then as long as it's in ok riding condition you should be fine to do the Katy Trail.
- the whole trail (now that the eastern end is in Machens, not St. Charles) is about 240 miles. How many miles/day do you want to do, and will you just go on the trail (or will you ride further, for instance to get to or from either KC or STL on either end)? Some would do the whole trail in two days, some would take a week.
- will you be camping or staying indoors?
- how much do you want to spend?
Easiest would be to get a big Carradice saddlebag and a big handlebar bag, and stay indoors. There are tons of places to stay on the trail or not far from it, they range in price from $5 for a night at the hostel in Tebbetts, to as expensive as you'd like in a place like Hermann or Jeff City.
If you want to camp, you'll probably need at least a rear rack. The Tivoli has fender eyelets on the dropouts, but you'd need to use p-clamps on the seat stays. Easy enough to do, you just need to make sure your rack/pannier combo gives you space for your feet while pedaling.
The crushed limestone surface of the Katy is pretty friendly to tires, as long as it's not really wet. I'd check tire clearance, as you'll probably want at least 28mm tires. Though 25s would probably work fine. I'm a fan of full fenders when riding too, especially with all of that dirt and dust along with possible wet - but that's up to you.
So, yeah, it's doable. Go cheaper on the bike and you'll spend more on lodging; get fully outfitted on the bike and you've got the option of camping.