My solution is inspired by this post.
You can't style a pseudo-class on a particular element alone, hence my below solution. Now, since you wanted inline JS, maybe you entertained using the onload() event for inline JS. However, onload only works on certain specialized HTML tags, such as <body>, <iframe>, etc, see here. Hence I am using it on the <body> tag. You can also not use inline CSS for pseudo-classes, see here. So, I am guessing this is as close as it gets for inline JS on pseudo-classes.
<!doctype html>
<html lang="en">
<body onload="document.styleSheets[0].insertRule('.firstLetterSpecial:first-letter { color: orange; }', 0);">
<div class="firstLetterSpecial" >
Hello Special 1
</div>
<div class="firstLetterSpecial" >
Hello Special 1
</div>
<div class="firstLetterSpecial" >
Hello Special 3
</div>
<div >
Hello Not special
</div>
</body>
</html>
Now, if you dont have access to the <body> tag, you can sneak in an unused and hidden <iframe>
<!doctype html>
<html lang="en">
<body>
<!-- I am being mean and sneak an iframe in here just to get the onload().-->
<iframe style="display:none;"
onload="document.styleSheets[0].insertRule('.firstLetterSpecial:first-letter { color: orange; }', 0);">
</iframe>
<div class="firstLetterSpecial" >
Hello Special 1
</div>
<div class="firstLetterSpecial" >
Hello Special 1
</div>
<div class="firstLetterSpecial" >
Hello Special 3
</div>
<div >
Hello Not special
</div>
</body>
</html>