I am playing the CD-ROM v 1.07 of TES: Arena.

I chose a High Elf Spellsword, and was sent to Firsthold after escaping the sewers. The only ones offering directions to a quest are the inkeepers. The king wants me to rescue his cousin in Garillon's Fortress. I can get through fine, it's when I get to the Ice Golem guarding the cousin is where I run into trouble. I have tried frontal assaults, trying to run in, grab her and run out, using fireball and Wizard Fire to attack it first (it doen't seem to even hit!) since Ice Golems are supposed to be vulnerable to fire attacks. I am using an Adamantium Short Sword, the only weapon better than Elven I could find in the city, but that thing keeps wiping my character out. I cast Orc's Strength, and Shield first before entering combat with it, but nothing seems to work. I've even tried to get it at a corner to keep it from hitting me, but it still gets me.

I tried searching for "Ice Golem" using the search function, but it gives me an error saying "one or all of search keywords are below four characters or I searched for words not allowed like "img, html" If anyone here has any suggestions on what to do, I would be most appreciative and grateful. If anyone needs more information, let me know. Thank you all!

Aside from the main quest to reclaim the Staff of Chaos, every other quest in Arena is completely randomized and there aren't any specific hints we can give you.
http://www.uesp.net/wiki/Arena:Monsters#Ice_Golem
This may help a little, then again, it may not. Weapons above elven should work against it-- Adamantium definitely qualifies. But your level may still be too low to score any decent hits. Damage from spells is also level based, so, maybe you're still too low a level right now.
Remember, the only quests that you don't have the option to fail are the main ones, the ones you're pointed to by the dreams Ria Silmane gives you. Failing another quest isn't the end of the world. Keep playing, and see what else happens?

The quest randomization wasn't the problem, but fighting the Ice Golem was, and you are probably right. I was going to try again using invisibility, and if that doesn't work, I think I have to go to another city/town, since the citizens aren't offering anything except for the inkeepers.

I did see the Ice Golem link on UESP Wiki, and I appreciate your response, MomentaryThought. Thank you very much for your time.

EDIT: I went to the shop to get an Adamantium War Axe, Invisibility spell and potions, and saved before accepting this quest again, to get a location near Firsthold. I got a different dungeon, and the Ice Golem (though not the first time) was caught on a corner, which allowed me to unleash on it until it fell. I finally succeeded. Thanks to MomentaryThought for your response again.

You're welcome! Hope other quests go smoother for you. Arena's got a steep curve when it comes to monsters-- if you don't already have one, get yourself a really good shield spell from the mage's guild. Don't take their off-the-shelf brand; design your own. A low base protection with a high per-level increase works out better in the long run. It'll be weaker at low levels, but at higher levels will hold up against far more damage.
Incidentally, in Arena it is only innkeepers and royalty that offer quests-- you won't ever get one from a merchant or temple unless it's part of the main quest (I seem to recall at least one of the main quest dungeons is forwarded to you through a temple). The quest engine isn't quite as diverse as Daggerfall. 90% of your quests will be offered in smoky tavern rooms and the few others by haughty nobility. Good luck.

Thank you!