Rome - yes, friend got shitfaced there with his "Give me wine!" sort of English, you'll be fine.
As a matter of fact, if your only desire to learn Italian comes from this trip, you'll only need to remember basic phrases like "Hello/Thanks/Bye/Sorry"
However, if you're keen on learning the language, duollingo will only manage to get you so far as to saying "My orange has fell from the table". It's not really practical unless you go in-hardcore and practice daily(If you do,
there are better ways of learning than duolingo. Here's a couple for you
The first one is probably the most excruciating(And oldest) one in the book, but brings results faster than the other.
Take an Italian book(or one which was translated into Italian) and find an English version. Then, put them side to side and start off by reading the italian one. Read 1-2 rows of the first page and then find the same two rows in the native-translated variant. As time goes by, you'll begin noticing patterns in sentences which will lead you to forming an understanding of how a language flows, its basic grammar and how to make small talk in it. Also, you can mix it up by adding in audio books while reading to get a grasp on pronunciation. Finally, if you keep on reading for about 2-4 hours each day, you should feel fluent by the end of week 6-7.
Alternatively you could "dive" into a language by almost completely omitting any factor in your life that has no (in your case) Italian present in it. For example:Movies, music, texts and any sort of chat(When you reach the appropriate level) should all be translated into the language you're attempting to learn. By doing this, you'll quickly proceed to understand what people are saying(Though how well you'll be able to reply is a different question).
You have just enough time to grasp Italian. A polyglot professor who lectured us on the first option learned Lithuanian in 6 weeks(as a native Russian, that is most impressive). For him it was a gruelling process of 6 hours per day, but it bore its fruit. Anyway, hope this helps. Good luck and spend your time wisely.
It won't harm if you do bring one. Then start learning the basic sentences so you'll know them without having to check the book everytime.
Reminds me of Mr. Bean in one of his movies where he had a phrase book, it was absolutely hilarious.
Seen the montypython one?