Nice quiz! :) Only the X-sound was a bit tricky. It's easy for me, but my mother tongue is with cyrillic letters, so... yeah. :)
I think it would be better if you change the name of the quiz to "Cyrillic alphabet", because this is the real name of the alphabet. In the description of the task you can mention, that the transliteration is from the Russian pronunciation of the letters. :)
I considered calling the quiz "Cyrillic Alphabet", but felt that the name wouldn't have been totally accurate since the quiz omits classical Cyrillic letters not found in modern Russian (such as Ѕ, which is still in use in Macedonian), and includes letters that are peculiar to Russian (such as Ы).
The only letter I missed was ы. I don't think this quiz is totally accurate. It is more of a "ih" sound, like in the word "bit." There is no real English equivalent. Also, tviordiy znahk (ъ) is not exactly silent. It is often referred to as a "hard sign." Myagkiy znahk (ь) is not silent either. It is referred to as a "soft sign." It makes the preceding letter have a softer sound, or lose its voice. You probably already knew that, though. Otherwise, very nice quiz!
The instructions didn't ask for English letters that sound the same. In many cases, as you say, there's no direct equivalent. The instructions asked for the standard transliteration scheme, such as might be used by international organizations. And such schemes almost uniformly represent "ы" as "y", for example in place names such as "Naberezhnye Chelny" (На́бережные Челны́).
I always thought that Й was similar to a Ñ in spanish, like in español where it take the normal sound of a И/N, and adds a y sound to the end. making an Й sound like ee-yuh. That's just how I learned it though, and that was a while ago so I might just be forgetting it.
I think it would be better if you change the name of the quiz to "Cyrillic alphabet", because this is the real name of the alphabet. In the description of the task you can mention, that the transliteration is from the Russian pronunciation of the letters. :)