Hello and welcome to the forum.
I got this tree as a gift for my birthday, and would really like to keep it alive. I would like some advice on the basics. I have read a lot about them, but am still not as confident as I would like to be.
Confidence will come with the passage of time, providing you don't kill your tree.
My tree does not have a draining hole, is this something I should fix.
Yes this is of the utmost importance. Frankly I'm at a loss to imagine a 25 year old tree without drainage. Can you describe it a little more, pictures will help here? Is there by any chance two pots, one inside the other? Can you describe the characteristics of the soil? Is it primarily dense and organic or loose and gritty. Is it homogeneous or is there two different types in the same pot, this may take some digging. Can the tree be slipped from its current container without the rootball falling apart. A slip pot would be the safest approach to take until you get some more experience, so a slightly larger pot should be located.
I think I would get nervous leaving it at home.
Not really sure what you mean by this. Are you keeping it at work? This species will be spending a large part of its life indoors in your climate so some supplemental lighting will probably be required as well.
The tree is happy right now, but I have not watered it yet or anything.
It may be OK right now but you are facing a critical time. It needs to be watered regularly of course but a pot without drainage is a disaster waiting to happen.
Here is a link that should help a bit, but your drainage issues must be addressed immediately.