Getting a game to age well largely depends on the art-style (so it never looks bad beyond resolution issues), and often requires inherently fun gameplay (so it doesn't feel gimmicky or dated). These are a huge part of what makes many classic games keep their places on the various "best games of all time" lists over the years, people can go back and play them without a care for how old the game is. Nostalgia isn't the only reason people love the classics, the fact they were damn good to begin with and age so gracefully is why people are still able to enjoy them.
As for the artstyle, it's generally a good idea to look at games which come out towards the end of a console's lifecycle or before a shift towards a different type of graphical style came about. People have figured out how to use the console's technical capabilities to make great artstyles for their games, and do so. Games which go for an absurd amount of detail, generally towards photorealistic, are quickly overshadowed by new tech; they also have a high tendency to look ugly, and high-resolution ugliness isn't a good thing unless you're going for horror. Even in the real world, artstyle matters. Go to Rome and look inside the many basilicas, and you'll know full well that there is an artstyle at work; ridiculously detailed at points, but there's always a theme going on.
The easiest example is the pixelated graphics from the late SNES era; at that point, pixels had more or less reached their limit for detail (technically continued to get finer on the GBA and a bit on the DS, but it was the end for the non-portable consoles) and the next consoles were going to use polygons and three-dimensional environments. No one can deny that games like Chrono Trigger and Final Fantasy VI look great despite using pixels; having two of the greatest soundtracks ever made for gaming (a claim which still stands to this day) doesn't hurt either.
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And now, for some (other) examples....
Super Metroid (SNES)
A classic among classics, this game is damned-near perfect. Expansiveness is often a hinderance for a game obtaining this lofty goal, keeping things sufficiently small is what will allow it. Super Metroid may be a tiny game by modern standards; but everything it does, it does to perfection. Riddled with many clever secrets as well; not hidden far away from the player, they're often in plain sight and just have to be approached from the right angle. The artstyle isn't exactly outstanding (looks good), but it is timeless.
Final Fantasy IX (PS1)
How often is a game built upon nostalgia, but doesn't rely on it to make it great? FFIX is an interesting anomaly in that regard; a throwback to the earlier era of the franchise with referrences galore, but also an outstanding game in its own right. Charming characters, a classical artstyle, and catchy tunes all over the place. Plenty of humour too, ranging from slapstick to even a few dirty jokes, "Oooh... soft.", that little scene says it all.
Metroid Prime (GC)
This game was actually something of a surprise; when it was released, people expected it to be a failure with it's many departures from it's predecessor. What happened was that Metroid Prime captured the feel of Super Metroid, effortlessly making the leap to three dimensions. The atmosphere was perfect, the gameplay sublime, and the graphics stunning. Speaking of which, the look of the game defied explanation; they were so richly detailed that one had to wonder if the little purple cube could handle it, but it did so while maintaining a rock-solid 60 frames per second. The game looked impossibly good (and still looks better than most games released today), and that's with no small thanks to it's artstyle.
