The longevity of indoor cats is twice that of indoor/outdoor cats. That should come as no surprise when you consider the many dangers that lurk outside: cat fights, dog attacks, car accidents, poisoning, infectious diseases, parasites etc'.
But the down side of not going out is boredom.

Cats are predators, they were meant to live an exciting and adventurous life. Though domestic cats sleep 18 to 19 hours a day, when they're awake they should keep busy hunting, playing and protecting their territory by looking out of windows, closely examining the closets and climbing to strategic view points. If the home environment doesn't provide enough physical and mental stimuli he might suffer from boredom. And boredom leads to a variety of behavior problems, such as aggression, scratching, crying, changes in appetite, sleepiness and even depression.

One of the most important things you can do for your cat is create a stimulating environment. An environment which offers a variety of activities cats love:
Windows to look out of
Interesting, changing views
Posts to scratch
Places to climb to
Toys to play with 

Window views
Create a nice window seat for your cat next to a window that offers an interesting view to the street, if possible. Choose a window to which access is easy and where your cat can lay undisturbed. 

Scratching posts
Put a number of scratching posts around the house. The post must be stable, wrapped in rough fabric such as sisal, and big enough to allow stretching.
Cats use scratching posts not only for sharpening their nails but mostly for territorial marking, both visually and by scent produced in tiny glands at the bottom of their paws.
Even a declawed cat will gladly use a scratching post; it's an instinctive and enjoyable behavior. 

A private hiding place
Cats like places where they feel safe. They will usually choose a high location, like the top of the dresser or the fridge. From there they can keep a close watch on their surroundings and watch for dangers. You can place a pillow or a little cardboard box in these spots. If you have enough space you can purchase a cat-furniture such as this:  

Toys
Find the right toys for your cat; different cats like different toys. The best toys are those which simulate hunting: a rod with colorful feathers dangling from it, a mouse on a string, soft balls which make noise, laser pointer-pens, etc'.

Other types of toys are those you can hide kitty-snacks in. Your cat will really have to work in order to get his treat. You may also consider serving the cat's usual food in unusual places which will force him to roam the house. He will enjoy the search and get a reward.
Once you've figured what's your cat's favorite toy it's best to buy a few of them since they tend to get misplaced (you might later discover them behind the sofa…).
Also, like most of us, cats lose interest and want novelties, so it's best to change between toys every couple of days, hide some and introduce others.

Try to devote 10 minutes twice a day for playing with your cat. That's a wonderful way to alleviate the boredom while strengthening the bond between you. With time, you are sure to see a happier, calmer cat.