I was just able to decipher enough of the Cyrillic inscription to figure out that it is from the Pochayiv (one of various spellings) Lavra, a monastery in Western Ukraine. There is a holy icon there, which is what is pictured on the cloth.
The photo was taken several years ago. I don't want to open up the case again to get at it and take a new picture.
If anyone is able to read this, I'd like to know what the rest of it says. I've tried putting all sorts of typical Blessed Virgin and devotional words and phrases into Google translate, to see if I can match any of it in Russian or Ukrainian.
http://365project.org/discuss/general/21735/can-anyone-help-translate-this#comment-641141
Ksenia @ksyu and Anna @nyusha have kindly provided me with translations.
I have to thank this wonderful 365 community of people for the help and encouragement given to others. This is just one example out of many, many.