I'd personally distrust the answers I got on such things from a stranger I didn't share a language w

We like to use a wireles router at home. We are looking for apartments the cheapest airline tickets in Italy, and some advertise broadband internet connection, but not wifi. Is our router we use here in the US compatible with internet the cheapest airline tickets connections in Italy? That way we would not be a limited in our options for apartments.
If you see an apartment you absolutely adore but it doesn't have wi-fi, tell the landlords you will rent the apartment if they will buy a router and hook it up and add it to the cost of the rental and they can keep the router. That would be about 100e out of your pocket, or less, but I think it would save you a lot of time and aggravation trying to set up your own system.
I'd strongly advise against your router idea. You've no way of knowing how the phone in the flat is connected (flats in Rome have all kinds of interfaces with the public network), or what contract the owner has with the phone company (the phone company might not be the ISP even when internet is offered). It's not at all universal for flats rented out even to have an accessible phone, never mind one with an internet contract.
I'd personally distrust the answers I got on such things from a stranger I didn't share a language with. If the wall wart (the voltage guide on your router) shows current incompatibilty, the transformer you'd have to bring with you will almost double the weight of your luggage (and add real cost). And the owner's permission doesn't mean the idea will work, or that your router will survive.
If they don't offer broadband access, you should assume there isn't any, and either rely on your mobile phone or take out a contract on a dongle. And even those solutions might not work inside many Roman buildings with thick walls. Holidays in Continental houses and flats often involve doing your email in a cafe or standing with your phone (or with your laptop and dongle) in an empty exposed space a few hundred yards away.
If a landlord is offering broadband access inside the apartment, you will have no problem using an Italian wi-fi router inside that apartment if the Italian router is set up for you inside the cheapest airline tickets the apartment, no matter how old the building or thick the walls. the cheapest airline tickets You don't have to go to an internet cafe or stand outside doing your e-mail in the part of the continent.
Rather than buy a chiavetta (dongle) the cheapest airline tickets that can only be used in Italy, consider investing in an iPad or some other form of tablet. the cheapest airline tickets That way, you can take the tablet home with you and have some use for it.
If for some reason you'd rather have a chiavetta that only works in Italy, you need to ask your landlord which phone company in the area offers the best coverage for that location. If you are renting that apartment for less than two years, you would do best to purchase a re-chargeable payment plan, not a monthly contract. In that plan, you pre-pay for so many minutes of access/GB, and when you need more, you get the chiavetta the cheapest airline tickets topped up at the local provider's store.
In case it wasn't clear, I was suggesting you buy an iPad or some other type of tablet in the US and bring that with you to Italy rather than a router. That way, you can take the device when you leave and have something you can use in the US. If you only need to check e-mail and the weather, or do searches for hotels, there are some types of e-book devices you can purchase in the US that will enable you to connect to the internet in Italy and perform those tasks. And a Kindle the cheapest airline tickets or some other brand of e-book is a nice thing to have both at home and abroad.
Also, I do own an IPad, but without 3G or 4 G, so it needs a wireless connection. We don't have a smart phone and haven't ever had a data connection for one, so I am not familiar with how they wouod work in Italy. We do have a tiny "MyFi" rounter which we used in HI, with a monthly plan from Verizon. I have no idea if that can be used in Italy or if we would need to re-register each time we move from one town to another.