By HS I'm guessing you mean Disney Hollywood Studio. You would have to take a bus to the EPCOT Theme park as they are completely separate parks. You would need park hopper tickets to get in if coming from another park in the same day. There is a boat over that drops you at a back entrance into the world showcase, but it is still considered a separate entry just like at the front gate.
As far as the monorail from MK, it will drop you at the main entrance to EPCOT. It does make a circular sweep through EPCOT as it a approaches, but only stops at the front.
The monorail while and excellent way to get around, only stops at the front entrance of MK, the front entrance of EPCOT, and there is a resort line that drops you at the entrance of several resorts: the Contemporary, the Polynesian, the rest escape me at the moment, but they are only the original MK resorts that were open when the monorail was first built over 20 years ago. There are always rumors that the Monorail will one day go to all the resorts and theme parks on the WDW property but not as of yet. It currently does not go to Disney Hollywood Studios, the Animal Kingdom, nor most of the resorts.
There is no magic walkway that leads from park to park as the park entrances and parking lots are all several miles apart from each other.
While I think that taking a break and heading over to EPCOT for some multicultural dining in your them park day is a great idea, it's not quite like hopping off a ride and walking over to the nearest restaurant. It takes a little planning and preparing.
Also as you said Future World is at the front of EPCOT with World Showcase in the back. If you made say a 2:00 Lunch reservation at a WS restaurant and were coming over the monorail from MK, I'd give myself a good 1hr to 1 1/2 hours to make it there. So you could exit MK between 12:30 and 1PM giving you time to catch the monorail, the ride itself, changing lines at the Ticket and Transportation center from the MK to the EPCOT line, getting off at EPCOT and walking back to the restaurant itself. You might save a little time with the boat that drops you at the back WS entrance, but only 15 - 30 min. with that route.
Also are you familiar with ADRs (Advanced Dining Reservations)? As you will need them to get into some restaurants, especially sit down table service, at EPCOT. A lot of people make theirs more than 6 months out which means a lot of the restaurants are booked full and you can't just walk in a get a table.