you're definitely not the only one who dreads being "that person"!!! i have a fear of being embarrassed by my child in public, too.
but what you have to remember, first, is that she's only 5 months old. 5 month olds don't throw temper tantrums, scream at the top of their lungs for the fun of watching mom squirm. they stay where you put them and (mostly) just sit there looking adorable. it will be MUCH easier now, than if she is 18 months or 2 years, and has never been to a restaurant. in fact, the more you take her out, and accustom her to the situation, the easier it will be.
and of course, as she gets older (closer to 2), she will have to learn what "no" is, and get used to hearing that.
BUT all of this is so far down the line, don't stress about it! there will be new sights, new people to interact with - it will only be an hour or so - i bet you anything she will be FINE! she's too little to offend people too badly - and the little bit of noise she makes, most people will be enchanted by, at her age, rather than irritated.
i would say, just for the record, that i would suggest to them that you all go to a family style restaurant, either like Applebees or maybe even a buffet like Ryans or Golden Corral (if you have those). i would avoid a stuffy "quiet" kind of place - but other than that, go for it! she will get lots of attention and she will be fine.
(oh also, you should make sure she is well fed, and has had a nap, if at all possible, so maybe you could suggest a time, based on her schedule, but i bet you already knew that.)
be brave, mama! :)