Are there organisms with fewer than 1000 neurons?
Short answer
As far as I know, a complete neural map (a connectome) is only available for the roundworm C. elegens, a nematode with only 302 neurons (fig. 1).


Fig. 1. C. elegans (left, size: ~1 mm) and connectome of C. elegans (right). sources: Utrecht University &amp; Farber (2012)

Background
Looking at the least complex of animals will be your best bet and nematodes (roundworms) like Caenorhabditis elegans are definitely a good option. C. elegans has some 300 neurons. Below is a schematic of phyla in Fig.2.

You mention insects; these critters are much more complex than roundworms. The total number of neurons varies with each insect, but for comparison: one of the lesser complex insects like the fruit fly Drosophila already has around 100k neurons, while a regular honey bee has about one million (source: Bio Teaching). 

Complexity of the organism is indeed an indicator of the number of neurons to be expected. Sponges, for instance (Fig. 1) have no neurons at all, so the least complex of animals won't help you. the next in line are the Cnidaria (Fig. 2). The Cnidaria include the jelly fish, and for example Hydra vulgaris has 5.6k neurons. 

So why do jelly fish feature more neurons? Because size also matters. Hydra vulgaris can grow up 15 mm, while C. elegans grows only up to 1 mm. See the wikipedia page for an informative list of #neurons in a host of species.  

A decent neuronal connectivity map (a connectome) only exists for C. elegans (Fig. 1) as far as I know, although other maps (Drosophila (Meinertzhagen, 2016) and human) are underway. 

References
- Farber, Sci Am February 2012
- Meinertzhagen, J Neurogenet (2016); 30(2): 62-8


Fig. 2. Phyla within the kingdom of animalia. source: Southwest Tennessee University College