most of this post is dependent on other things and cant really be answered based on the info tht you have given.
What do you consider the carribean coast? For most people that means the bluefields/corn islands area. And there are no roads connecting the pacific side to the carribean. So you take a 6 hour bus to rama, and then take a boat up the river for a few more hours to bluefields. From there you can take different small boats to pearl lagoon, kukra hill, corn islands or the surrounding areas.
Alternatively you can fly there to bluefields from managua round trip for less than 100 bucks, or round trip to corn islands for less than 150.
There is also the northern carribean coast, puerto cabezas, which is much less traveled. There is a road there that isnt in great condition, so its a lengthy 12+ hour bus trip from Managua. Less if you are already on the way, like Matagalpa and catch a passing bus (youll have to ask around in the town when the buses pass through as they arent super frequent).
As for having enough time, it depends. The carribean coast is not a day trip, but if you fly, or even take land travel, you could see it in just a few days. If you like to take things slower and experience the culture, and see multiple places, or do up river and see native villages, etc. You will want at least a week for that stuff.
As for fitting that in with your other things, thats also dependent on what you want to do there. Some people see Leon in a day trip, others can spend weeks there. Same with Granada. So it all depends on what you want to do, or what kind of trip you would like to have.
Also, unless you are going to Corn Islands, or going to really rough it and try to go to cayos miskitos, etc. You may wanna look up info about the carribean beaches of Nicaragua. They probably arent quite what you are thinking of in your mind. Most of the coast is undeveloped jungle and very muddy, etc. Not much for white sand, teal water, super clean areas.
If you want beach experiences, the pacific has miles of open empty beaches. The waters not as calm, but its a beach experience.