Mosquitos (called zancudos locally) and insects in general can be abundant in Nicaragua; certainly Managua is no different in that respect. However, it is not like some places in North America where a veritable cloud of insects forms around you anytime you slow down or stop moving. That's not to say that the bugs are not an issue at all, they definitely can be, and you should plan on taking precautions. You definitely will want to prevent mosquito bites, as they carry serious diseases like dengue and malaria.
The insects are much more prevalent in the rainy season from May-October. Zancudos flourish anywhere there are puddles, marshy areas, rice fields, etc. The mosquitoes are also a problem in some urban areas that have stagnant water in discarded items like old tires where the insects can lay eggs and hatch larvae.
The species of mosquitoes that are active in the evening and night can carry malaria, while those that are active in the daytime can carry dengue. Neither of which you will want any part of, take my word for it!
The name of the game is prevention, as in cover up your arms and legs with long sleeves and long pants to stop the mosquitos from being able to bite at your exposed skin. This is especially important when the mosquitoes are more active, dusk and dawn mostly. Proper clothing and avoiding being outside at the peak times works much better then any bug spray.
The other preventive measures have to do with your lodging. The best situation is to have a room with screened windows and doors, the next best is to sleep using a mosquito net. It won't hurt to do both if you are really concerned. Directing a fan to blow insects away from your body as you sleep is also a good idea. Some people use mosquito spirals as a sort of an incense to keep away the bugs, but in my experience, they just release a toxic smoke that causes more harm to the individuals than the insects.
As far as other insects, the big two that people worry about (for good reason...) are spiders and scorpions. These too exist in relative abundance in any tropical country and Nicaragua is no exception. Spiders generally do not seek out human victims, and an easy way to avoid them is to make sure to shake out your shoes before you put them on. Scorpions are a bit more of a worry as they tend to hide out in dark places, woodpiles, and other forgotten corners. The bites can be serious for those subject to anaphylactic shock. For the rest of us, a slight numbing of the lips and tongue might occur.
The other insect that is nasty, but you are very unlikely to be in contact with, is the Chagas bug, locally known as chinche. They are usually found in very unkempt and falling down buildings, usually made of adobe. These bugs climb on your face, bite you, then, (and here is the nasty part...) they defecate in the wound. A small percentage of those bitten get the Chagas disease, which involves swelling, fever, and a reappearance in a few years. The way to prevent this is to not stay in old crumbling adobe houses.
So, aren't you glad you asked?