That is typically where a rapidfire shifter will fail, the mechanism on the larger push lever is very often the first thing to go. You might want to try opening it to see if the mechanism is sticky or dirty, sometimes the pawls are simply failing to engage. You can then clean and lubricate it from there, which is pretty easy if that's all the trouble is. It often is.

The cable can also potentially be a trouble if it is kinked or frayed inside where you can't see it, although this is less common.

Past that, other types of service to rapidfire shifters are limited.