It does not correspond to any English sound in particular. You may start with "i" in "hit" and move your tongue a bit to the back (about as mid-way as e in "her")
If you are familiar with Russian "Ы", it is somewhat similar, but NOT the same. The English "i" in "hit" is a good model (only don't turn it into "ea" in "meat", which is Ukranian і). Maybe, move your tongue a bit to the back. However, if you go further back, you'll get Russian "ы", which will not sound especially native to a Ukranian speaker :)
I thought it sounded like "bI", my ears didn't get the difference.
About tyt, I got a russian textbook from the 1960s, and I've learned from it words like pero (pen) and tut (here). I took some classes last year and apparently those words in current russian sound a bit odd. The teacher prefers zdes' for here, for example, and I forgot the word for pen, she told pero was like feather, ie, would sound like an archaic word.
Finally I meet тут again! Waiting for the word for pen in Ukrainian :)
"перо" is a "feather" and the word used for nib and pens that have it ( not exactly common these days).
A graphic tablets' pen or slylus is "перо", too. It is also a metaphor used when attributing a literary work to a writer (we say that it "belongs to his feather) — which is understandable because goose feathers were a popular material to make quills.
A modern ballpoint pen is "ручка" both in Russian and Ukrainian.