ETA: After reading your "So What Happened," I see that you don't have a wide selection of items and stores near you. That's fine. But I don't see why you expect a 2 year old to hold a spoon or do everything himself. What you do is feed him. Does it take time? Yes. But that's bonding time for you two, and it's how he learns to use a spoon, a fork, and other utensils. Watching you is how he learns to eat new foods. Holding a small regular cup and helping him is how he will learn to use that. You don't really need a sippy cup anyway - just a small cup so he doesn't spill too much.
What I see from all your posts is that you have unrealistic expectations of a young child. You expect him to learn everything immediately and at a very young age. You don't seem to have enough experience or enough experienced parents around you to give you perspective on what's age-appropriate, what a reasonable learning curve is, and what the reasons are for kids not doing things right away. Your prior posts and this one seem to blame personality and moods rather than inexperience and youth. I think that's going to hinder you long term, and make you very unhappy as a parent. I really hope you get some books on normal stages of development, the range of what's appropriate (like some kids potty train at 2, most at 3 or more, and quite a few after 4). They're people and individuals, not machines you can program or adult animals you can train.
Original answer:
There are lots of ways to get liquid into a kid. Popsicles, jello, soup, watery fruits like watermelon, and more.
And why not give him a cup he can use? If he's chewing on the old cup, you can try a travel coffee cup with a small hole and put a straw in it. You don't want to use plastic straws because they are so incredibly dangerous to the environment and animals that ingest them, but you can use one for a few days while you order paper straws or stainless steel ones (not chewable!). Kids who like to suck on things can use them sometimes better than sippy cups. If he's not having bowel movements, you need to increase fluids and fiber rather than fight the battle of the cup. In the short run, take him off all dairy (including cheese), bananas, rice etc. and increase fruits and vegetables. If he's not on whole grains, he needs to be.
If he doesn't have a bowel movement in another day, call the pediatrician and discuss what you can safely give him to help. Sometimes a stool softener or suppository works, but talk to the doctor first.