The vast majority of software development and even computer science relies mostly on logic rather than "numeric" math (some sub-disciplines aside).
I initially ignored CS because I hated math in high school and thought it would be "too much math". Turns out I am really good at CS even though I sucked at things like calculus.
So my first point would be to make sure you try some actual programming before you get scared away about any "math"-related concerns.
Secondly:
> Is the analytical part of your brain something you can build and improve?
Absolutely. It is just like a muscle. The more you practice, the easier it will become. To be able to build powerful software you have to learn new patterns of thinking, which takes time. But it is definitely doable, provided you enjoy it enough to push through the tedium. But it's not everyone's cup of tea.