Why does the hospital in Another have no fourth floor? In the hospital where Sakakibara was having his checkups and where Sanae worked, the elevator had no 4th floor.

Now, I know in Japanese culture the numbers 4 (四 shi) and 7 (七 shichi) have "shi" in them which is "death" (死 shi) and the Japanese are very superstitious, so they have alternate names for 4 and 7 (yon and nana).

So I'm wondering, is it just in Another, where the theme is death, that the 4th floor (and possibly the 7th floor, never really noticed) is missing, or is it a cultural thing in Japan to label the 4th floor as the 5th floor?
This is true for many East Asian cultures, because, as you pointed out, 4 sounds like "Death". You can read about it here. It's the same as the idea that many western hotels skip the 13th floor and instead label it the 14th floor because 13 is a superstitious number.

Japanese hotels sometimes also skip the 13th floor.