I think the shelf life is fairly long on those batteries, and IIRC Noise plays his DC often, so the battery should be replaced regardless. But it's worth a shot in the mean time. If you can get it to hold a charge and then play it a few times a day every day, it might be able to keep the saves for a little while longer. But yeah, even if so, you'd wanna replace it anyway.
Noise, have you ever seen examples where people have used rechargeable AA or AAA batteries as a replacement? The two batteries makes 3 volts, which is what the Dreamcast battery is, so it works just as well. I'm not suggesting to do this, but I just thought I'd mention it.
Looks like the Dreamcast battery is an ML2020. I've seen that from multiple sources, but it's always a good idea to double check. Just open your DC and take a look at the battery (it's right in front, you just have to pop the top off) and make sure it's an ML2020.
A lot of people seem to solder in a coin battery holder with two wires, which I guess makes changing the battery down the road easier. But if you ask me, getting a battery with the solder tabs on it makes it much easier. Especially for a novice. So this is what I would suggest you do.
It's an ML2020, and apparently the "G1AN" is the spacing for the solder tabs, so it'll be a direct replacement for the Dreamcast (the 3 legs will line up perfectly). DigiKey stocks them, and for only $2.55, but it might be cheaper to get one off eBay. I see this one on eBay for $6.95 shipped, so it might be the same price actually. Looks like the same exact battery from DigiKey either way.
Looking at the tabbed battery, it's keyed (the positive side has two legs and the negative only has one) so you wont be able to stick it in the wrong way.
Here's my tips for a novice:
Use the guide linked above. When you get to the desoldering part, it'll be easier to cut the legs first (with a dike or small wires cutter). The positive side of the battery that has the two legs, cut those legs. Use the soldering iron to melt the solder on the one negative leg and pull out the battery. Now with the two (cut) positive legs still soldered in, you can remove them one at a time. Just melt them one at a time and use your dikes as pliers to pull them out. I say do it this way because it's a lot easier than trying to remove the battery with the three legs soldered in. Of course, if you had a desoldering iron or braid or solder sucker or whatever, you could just do it that way. But I'm assuming you don't have such things.
Now that it's out, just plop the new battery in and solder the three legs. First, watch this video:
That video shows all you need to know, but here's the break down: When you put the new battery in, bend the legs slightly on the bottom of the board. This'll hold the battery in place while you're soldering so you don't need a third arm. With your soldering iron heated up, assuming the tip is clean, tin the tip. You wanna position the tip so that it's touching both the leg and the solder pad. Heat up the leg and solder pad with the iron for a few seconds, apply some solder, then remove the iron. Repeat for the next two legs and your done.