Residence Inn is on Gay Street downtown. It's not right in the Short North, but it's not too far away. The Hampton Inn at 501 N. High would be the closest hotel to the Short North. The hotels along Nationwide Blvd. near the arena would also be more handy to the SN.
Definitely there is more active street life in the Short North, and a lot more shopping. Downtown is fairly dead outside of business hours (although Gay Street is one of the places with some activity in the evening). If you don't have a particular reason to stay right downtown, I would stay closer to the Short North.
If you do stay at the Residence Inn, you'll be right across the street from the Tip Top, a bar & restaurant which is very good, and has vegetarian options (although I must admit that their pot roast sandwich is one of the things that makes me want to rethink my commitment to vegetarianism...). Due Amici next door to Tip Top is popular with a lot of people, although I've honestly found it to be kind of 'meh.' In Pearl Alley, just south of Gay, El Arepazo is a Venezuelan restaurant open for lunch where you can find some good veggie options -- the vegetarian patacon is very good. A couple blocks south of Gay at State & Third, Market 65 serves very good salads and wraps. I believe they are open for dinner hours. It's a very popular lunch spot. Latitude 41 in the Renaissance Hotel on 3rd gets kind of mixed reviews. I liked it okay the one time I ate there, although I was still eating meat then.
If you stay at the Hampton, you'll have many, many options. You'll be right across the street from the North Market, which has a lot of different food options. Northstar Cafe at Second & High is veg-friendly. The veggie burger is very good there. I also like the basil burrito with tofu. Betty's and Surly Girl (owned by the same group as the Tip Top downtown) generally have good vegetarian options. Nida's Thai on High is good and has good vegetarian choices.
A couple of spots that you would need to drive to, but that might be worth checking out -- Angry Baker on Oak in Old Towne East for breakfast or lunch. They have a new black bean burger on the menu that is excellent. And really, everything I've had there has been good. Explorer's Club on South High in Merion Village is a pretty new restaurant with some good vegetarian options. I've liked everything I've had there. They're open for lunch and dinner and weekend brunch. Katalina's Cafe Corner at Third & Pennsylvania in Harrison West has really good sandwiches for breakfast or lunch (you could walk here from the Short North, although it would be a good hike). With only one day in town, you'll probably find enough to do in the Short North without traveling to one of these spots, but they are good, so just putting them out there as options.