Cat whole his life from young age was living on a first floor, so he easily was going for a walk on himself and then knew where and when to go back, now we moved to another apartment on 8th floor, in different city part and we fear to let him out because as we think he won't return, but it's clearly seen what it's hard for him to live without going outdoor, so we think to buy a leash and just to take him for a walk we us, but question is will it help him, or tease more?
I have had two cats that were leashed trained, as we moved to areas not safe for them to go out on their own. They learned quickly that if they wanted to go out it was on a leash, and they adjusted to it. It did NOT tease them. Depending on the temperment of your cat, you could leash train it.
First benefit of leash training is that you spend time with your cat. More time, more attention, is always beneficial to any pet and its relationship with you.
Start slowly. If on the 8th floor, for the first walk you may stay on the 8th floor while your cat explores. Let the cat explore at his own pace, don't force the cat to go farther than it's comfortable going. If the cat is fearful, scoop it up and go directly back to your apartment. Equate safety with your apartment. Don't force the cat to go for a walk. Make it a positive experience only.
Be consistant. Make putting on the collar or attaching the leash always lead to being allowed to go outside of the apartment. No collar / leash, no out. Make the walk at the same time each day. Set an alarm (different from your wake-up alarm) to signal walk / play time. This will make it less likely that the cat will howl at night to go outside, or wake you up to play. It will take some time, but the cat will learn these signals and cues.
Start the walk at 10 minutes. Slowly expand the walk to however long you can reasonably do it. This is up to you. If you only have 30 minutes each day, then again, be consistent with it. Getting out of the building might take up most of the time, but it's time well spent together.
Don't try to walk the cat like you would a dog. The walk will be a slow walk, with many stops, as you walk at the cat's pace. Don't pull the cat along. Cats can slip collars, even harnesses, by pulling back while you pull forward.
I'd use the stairs and NOT the elevator. Elevators can freak cats out. Stairs are better for exercise, at first, the stairs may be the entire walk. Stairs help release the pent up energy in your cat although taking the stairs back up to the eighth floor will probably be harder on you.
Never drop the leash. A cat can freak out and run into traffic, or the leash could get caught on something. The walk is not for you, the walk is for the cat. You don't run to the store with the cat on a leash. Keep your needs separate from the cat's walk. This time is just for the cat. You may end up standing in one spot for a while as the cat hunches down to watch things. This is ok. This is what a cat walk is sometimes.
See if there's a park nearby. (Not a dog park.) It might take a while, or you may want to carry him, but getting him to a safe destination where he can explore would be a good idea. If using a carrier, put him in it with the leash. Don't take the leash off until you are back in your apartment.
If other animals / children / people approach, step between your animal and the other animal / child / person. This signals to your cat that you stand to protect them. Once between, then if necessary, you can pick your cat up but don't hold the cat and talk to people. If still on the ground and the cat decides to approach the other person, that's the cats decision, and if you feel it's safe, let them. Don't let dogs sniff or get near. Be very watchful if a child goes near the cat. Take your cues from your cat.
If the weather is bad and you've gotten to the point of going outside the building, start the walk and let the cat go down and out the building so that they can see that they really don't want to go out. They will always look for the door into summer. The adventure will be going to the door and going back up. It's still "out" to them and better than nothing.
When the walk is over, pick the cat up and go directly back to your apartment. Make sure to hold the cat securely so the cat feels safe. Don't one arm it. Support the rear haunches and gently but firmly, hold the cat to your body with the other hand. The cat may protest but let him know you're the boss and the one in control. Don't brook any nonsense.
Be careful when using the keys to open the door. Put the cat down beside you if you can and let them walk in the building on leash. If he doesn't go right inside then pick him up, get in the building and close the door. You may carry him directly to your apartment, or continue the walk inside the building until you both get back to your apartment. Again, put the cat down and let him walk back inside.
After the walk, make a big fuss and praise him.
If at any point the cat becomes fearful, get the cat home right away.
And if all of this seems too much, then invest in some pet toys and play with your cat for an hour or so everyday instead of walking them. This will release the pent up energy, provide quality time for you and your cat, and make your pet very happy. The cat may still look out the window and wish for out, but if you can exhaust them with play, their life will still be good.
You could also alternate walks with indoor play time and/or combine them. If you make Indoor Play time really good, the cat may not even want to go out.