You may want to look into "ART", Active Release Technique. It's very effective for increasing soft tissue health, which may be your problem. They usually have you go for 4+ sessions, but the initial results should be noticeable.
The headstand will not get you to performing a handstand. Yes, you will get used to being inverted, but there are other, better ways to do that. Don't get me wrong, the headstand is a good, simple position to control before the 'real deal' (handstand), but the balancing mechanics are very different.
Walking on your hands is easier than standing still on your hands. If you want to get good at handbalancing (in any capacity, not just handstands), you need to spend a significant amount of time practicing. Also, you need to address joint preparation, mobility work, and body line drills otherwise you won't get far. This is SKILL work, not strength training, and training daily (or as close to daily as possible) will provide the most benefit. The handstand is surprisingly very technical and has many nuances, despite the simple appearance. If you want to get good at balancing on your hands while stationary, I suggest you find a good handbalancing coach. There are a few good starting points on the internet, but they don't cover the subject in much depth.
