No one else has mentioned the "w" word, so I will: weather. June is not July or August, but it still may be quite hot in Rome while you're there (what are the exact dates?). You'll find that air conditioning is not as prevalent in Rome as in US cities. Small bars, cafes, restaurants and shops often do not have a/c. (Hint: If the door is left open on a hot day, there is no a/c!) Some of the museums aren't air conditioned either. And of course there's lots of outdoor sightseeing in Rome.
Wunderground provides day-by-day weather stats for most major cities going back 15 or 20 years. I've linked to the June 2017 data for Rome. You can use the pull-down box to change the date to 2016, 2015, etc., to get an idea of the range of temperatures you might experience. I'd be less nervous about early June than about late June, but it's still a plus to be within convenient walking distance of the core of the historic area.
If you are unlucky with the weather, you'll be very glad to be staying somewhere that's an easy commute from your sightseeing targets so you have a place to cool down from the mid-day heat. That could be by walking (I'd consider a 60-minute extra roundtrip in the middle of the day to be a little long) or by bus or subway. If there's a bus from your area that heads to the historic center, that will work.
I assume that the apartment you've booked is air conditioned, but do verify that. Many European apartments are not, but I think it would be unusual for a rental unit in Rome not to have a/c.
I have a strong preference for being in a neighborhood with multiple restaurant options in my (casual/moderate) price range. I have developed that preference as a result of staying in hotels that were convenient to the train station but very skimpy on places to eat. This may not be an issue for you, depending on how you manage your days and how you feel about needing a bus to get back to your apartment when you finish dinner (probably after 8 PM). If the apartment is in a middle-class neighborhood equipped with eating establishments frequented by Romans, that is great.