The short answer is no. EDIT: Actually the short answer is yes, see below. Opacity acts like a multiplier on any rgba values present within the same element. Consider this:
div {
background-color:rgba(0,0,0,0.5);
opacity:0.5;
}
The background opacity of 0.5 (the alpha value) effectively* becomes 0.25 when multiplied by the opacity: 0.5 x 0.5 = 0.25. See this fiddle.
*I say effectively because the colours are slightly different, but it's virtually impossible to tell with the naked eye.
EDIT: As @vals pointed out, you can get the effect originally asked for:
#tagholder:hover {
background: rgba(255, 0, 0, 0.25) url("../images/arrow-white.png");
background-repeat:no-repeat;
background-position: center;
opacity: 0.8;
}
This would give the whole element an overall opacity of 0.8, while the background colour would have an opacity of 0.2 (0.8 * 0.25 = 0.2). Note, this would also affect the opacity of borders and text colour as well.