Well, here's a somewhat different answer for you to ponder! Rosemary is a mediterranean herb - it grows well in fairly poor, free draining soil in full sun. Yours definitely needs a larger pot, and I'd suggest you prune back the mostly dead parts at the time of repotting. I would not, though, recommend any further fertilizer - this plant has meagre fertilizer needs, and you've been giving it fertilizer probably too often already, plus the new potting soil you use will have some small amount of feed within it which will be sufficient ongoing.
These are medium sized shrubs in the ground, reaching 6 x 6 feet or more - it will not be a permanent plant in a pot, there will come a point where it will outgrow any pot you provide for it, although you will likely have it for up to 5 years, maybe more if you provide bigger pots ongoing.
Fertilize in spring with a slow release granular balanced fertilizer (NPK something like 7-7-7) and; you can either leave it at that till the following spring, or repeat the fertilizer 6 weeks later, depending on the recommendations of the product you choose, but keep it watered as necessary, allowing the pot to drain down freely to get rid of excess water. Raising the pot off the ground by sitting it on something (pot feet, bits of slate or whatever) will help with drainage. Note that Rosemary hates wet feet in winter, so if you have very wet winters, that might be an issue.
In terms of insect infestation, Rosemary oil is a natural insecticide so the plant is largely untroubled by most insects, other than Rosemary Beetle and occasionally a mite infestation, so you do not need to apply neem as a preventative measure.