Yes, I would say so, in the ultimate sense it will go away the closer one gets to Nibbana. But there are many intensities of a fear one is experiencing, because a fear is an aversion, that “something should not happen that would go against my hope/wish”, and so as with all other defilements, it is always important to be mindful of the emotion, understand the underlying attachment or craving, reflect on this with the understanding of anicca/dukkha/anatta (e.g. reflecting about “what exactly is it one is afraid of?” “why do I hold as true or important for myself, that it makes me afraid to lose or face such a situation?”, and with the understanding that holding on and “trying to manage” any kind of rupa is stressful, subjects one to suffering, one will be able to handle the emotion and reach to deeper understanding of oneself.
One other aspect is the simple truth that any rupa one is attached to, if one does not have it – that makes one grief for it, desire it, want it, long for it, … and if one has it, one will be afraid to lose it (hence fear is part of every desire/craving/attachment).
But this is only a general idea, I think that fears can have many different forms, e.g. fear from an enemy could be very much a problem if one is engaged in many immoral activities or if one is acting in a way that makes it easy for you to be an enemy. Such a kind of fear can easily be changed once one understands some Dhamma and progresses on the path. On the other hand some fears are more “self-preservatory”, as the latter you mentioned, fear from an insect, where one is “afraid to lose one’s life” or be hurt, and these ones might only really go away with Arahant stage, because they are more related to the mana- cetasika (sense of pride, sense of self).
That would be my ideas.