Is age bad for solid state amps?

1998 is not old for an amplifier. Caps have a lifespan of 20 years or so. (small ones can fail at anytime, but the big ones usually last 20 years or so)
The ARC has a great reputation for quality, and for doing good repair work. I would not hesitate to buy an onlder Audio Research product if it was what i wanted.
(I do own an Audio Research Sp-15 preamp, and an Audio Research phono PH-2)

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.

Electrolytic caps can last 50 years but every 10degC in temperature increase cuts life by half (it is simply drying out). It is gradual process that increases ESR (Effective Series Resistance) of capacitor up to a point where ESR is so high that capacitor is getting hot from the power losses. Thermal runaway is even possible since hot capacitor has even higher ESR that heats up capacitor further creating condition for explosion (capacitors have vent/fuse).

I would replace 10 year old cap only if it was exposed to temperature (next to tube, etc). Be careful with amplifier that was in storage for a long time (many years) since electrolyte eats up aluminium oxide that serves as isolation (lowering breakdown voltage). Voltage on capacitor builds up aluminum oxide layer again - increasing breakdown voltage but it has to be done slowly - over few days using variable transformer slowly increasing voltage in steps.