Battery Ampere-hour rating vs Battery Amps (not an experienced user)

Your hunch that batteries have a current limitation is correct. In general, it's hard to tell the current rating [A] from capacity [A·h]. You have to look it up in the datasheet. A lot depends on the design of the battery.

For example: coin cells with 500mAh capacity may have only 3mA max current.
Another (opposite) example: automotive starter battery with 40Ah capacity may have 500A max current.

Lead-acid batteries are interesting in this respect, because there are two distinct types.

1. Starter lead-acid batteries are designed specifically to deliver high peak current for a short period of time. Deep discharge, however, dramatically shortens the life of a starter battery. So, it's not suited for routine operation at high depths of discharge. Your typical starter battery in the automobile works at very shallow depth of discharge.
2. Deep cycle lead-acid batteries are designed (as name suggests) to discharge further. But they can not provide as much instantaneous current.
   Here's an example datasheet for a deep cycle battery. Have a look at the nominal capacity on p.1. Notice that capacity depends on discharge current (i.e. the rate of discharge).

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Depth of Discharge    Starter Battery    Deep-cycle Battery

100%                  12–15 cycles       150–200 cycles

50%                   100–120 cycles     400–500 cycles

30%                   130–150 cycles     1,000 and more cycles

(Source.)

p.s. If you want to read-up, here's an excellent web site on batteries - Battery University.

No, not necessarily.

Amp-hours tells you how much energy is in the battery, the amps tells you how quickly that energy can be delivered (For the most part, for a fixed voltage).

Your suspicion is well founded, because the Amp-hour doesn't tell the story on the batteries wattage capability.

Car batteries post their peak amperage because that is important when starting a vehicle. For your purposes, you'll want to find a deep cycle battery. Significantly discharging a car battery pretty much ruins it. Deep cycle batteries are designed to be discharged to a much greater degree, down to as low as 20%, and those are rated such that they can deliver the rated current continuously.

You may be able to find a suitable battery, depending on how long you are going to run your microwave. A decent 20+ amp rated deep cycle should still be able to unload 150 amps for a minute or two. Don't run the battery full blast non-stop though, it will overheat. Shop around for batteries for boats, RVs, etc to find good batteries.