Is age bad for solid state amps?

Yes, but probably not a worry for the age of the amps your considering, and they usually can be refurbished, especially with an established brand like ARC.

Well, my Nakamichi deck is 30 years old, I bought it 10 years ago and heard no difference until I had it cleaned and aligned. Electronics seems to be just fine. Also, I bought Audiolab integrated new 12 years ago and again - no difference.
So it depends, I guess.

The above comments are generally correct, as is the comment that an amp made in 1998 is not old.

I would add, however, that age with solid-state amps is more of an issue than it is with tube amps. The output transistors in solid-state amps can go out of production, and if such an amp loses an output device, it can't be fixed and the amp is toast. This won't happen to a tube amp that uses output tubes common in the tens of thousands of tubed guitar amps that are sold each year, e.g., EL-34's, 6L6, 6V6, 6550, etc., or that use a classic triode like the 300B, i.e., this won't happen with most tube amps.

In addition, when a solid-state amp breaks, it usually has to be sent to a tech or to the manufacturer to be repaired. When a tube amp malfunctions, it's more often than not a tube that is the source of the problem, and they are user-replaceable.

Finally, with a tube amp, the tubes are the circuit, and when the amp is retubed, it's basically a brand-new new amp.

I remember I read an interview with Jeff Rowland somewhere. He promised that he would fix every amplifier he'd ever made, for as long as he was alive. Maybe ARC will too.
It is their problem to find proper transistors and other stuff not ours.
True, well-made and maintained tube amps can probably last for longer than hundred years, no one knows yet.