Is an 150 psi air compressor enough for spray staining or painting?
Probably not, because 150 psi air compressor labels are generally found on smaller pancake and hot-dog type models. The important number is the volume of air supplied which is labelled in cubic feet per minute. (CFM or SCFM on the compressor label) Think of it this way, a bicycle pump provides a very high pressure but you wouldn't want to try spraying paint with one!

Typically spray guns require 6-10 CFM while small portables put out 3-4ish. Some of them are embarrassingly close to the low end too. I vaguely recall seeing one model that was rated below 2 CFM @ 40psi. It's hard to make suggestions without knowing your use case, but here are a couple of options:

Small Compressor, Small Gun

Detail guns need less air. The current HVLP touchup gun at Harbor Freight requires 3.2 to 5.6 CFM @ 40 PSI. There are portable compressors that can meet that output. Of course you won't want to paint a house or car with it, but it is an option for the occasional small piece.

Small Compressor, Special Tools

It's not my forte, but you can run conversion guns and pressure pots off the next step up in compressors. Price wise it's probably the worst of all worlds unless you already have a serious portable compressor.

HVLP Turbine

If you haven't noticed, the compressor is getting to be pretty expensive. Tool companies have noticed and now offer portable turbines for spray guns. They're a reasonable solution if you don't need pneumatic tools other than paint guns. 

Airless Sprayers

These pump paint through a tip to aerosolize it, and can often be rented from paint stores. The big draw is that they'll shoot latex right from the can without complaint. It's really the way to go for paints and large surfaces.

EDIT - Specific to your setup

Your compressor matches your gun requirements exactly. Unfortunately most compressors have a duty cycle which limits them to running part time to prevent overheating. Start by checking the manual to find the duty cycle and see if it is an issue. It may not be an issue if you're limited to 7 out of every 10 minutes if you're not painting a building.

If you're doing a lot of painting then a paint pot may be a reasonable purchase. (Alternatively an additional air tank if you use a lot of pneumatic tools or small amounts of paint.) It wouldn't be the most cost effective new purchase, but it would be cheaper than a new compressor. It would also remain useful should you ever upgrade the compressor.