Our Neighborhood bought a new metal bridge to replace the old one but the new metal bridge wasn't primed and there is rust all over it.
Is there a product(s) or not too expensive way to paint the metal without having to sand off all the rust?

no such product exists. Although you can find such things at the big box stores that claim to eat rust. They simpy do not work. If they did you would not see the bridge painters on the highways sand blasting the metal bridges they would simply scrape and paint. The only true and tride method of rust removal and painting is to sand or sand-blast or soda-blast the metal then properly clean with solvent and paint with good quality primer for metal.
I would check the local rental center and rent a sand blaster with compressor if you want the paint to last for any amount of time.

A lot of what home care says is true, but highway bridges are expensive and disruptive to repaint and often go for years between refinishes. Therefore they tend to get in pretty bad condition before they get attention. I suspect that being a new structure, the rust on your bridge is mainly on the surface and not scaley, deep rust. You may need only to wirebrush the surface and apply a good quality primer and finish coat. I would contact a local paint store such as Sherwin-Williams or the like and ask to have their local tech rep visit the site and make a recommendation. It will be far cheaper to coat it right the first time, than attempt to blast clean it later, especially if it is over a stream or someplace where blasting comes with an environmental impact study, etc. Environmental Protection Agencies tend to get real itchy about sand blasting residue entering the ground, water, air, etc., particularly if the residue contains old paint.

Thanks.
I've found some coatings touted on the web that go over the rust and interact with it to form a strong coat that keeps the rust from progressing. Since our metal doesn't have any flakes just a dust like coating it wouldn't need much prep. Just wipe off the loose rust and brush or spray on the product. Some products don't even need to be painted over. They range in price from $40-125/gal. Seems like the savings in prep time would justify their higher cost. The expensive ones don't need to be painted over saving more time but colors are limited.
Has anyone had any experience with any of these products (POR-15, Rust Doctor, Rust Bullet)?

You did not read my post. There is no such thing tht turns the rust by a special process into a magic coating that can be painted. The rust is the result of decay of the surface of the metal. This decay cannot be converted into a coating.
Think of it this way. Picture the rust under a microscope. It would look like a gravel road. The gravel being the rust particals coating the rest of the metal under it. Now take a bucket of paint, tar or any thick liquid. Pour it onto the gravel. Looks good. covers the gravel and interacts with the stone to form a strong coat.
After a while with cars driving over it. or in your case the heat and cooling seasons. the gravel will begin to crack and become loose. Much like the dusty rust on the steel will do. The only diffrence being you will spend a LOT more time scrapeing and re-painting the bridge in a very short while. By then the paint company that sold you the coating is either out of business or, will not warrant the material because of some special condition in their warranty.
I suggest that you NEVER purchase a unknown product from the internet. If these products were as good as they claim you would see them in the higher quality paint stores or big box stores.
Take the time to do the job right the first time. Weather you sand or blast the finish off, and Maintenance 6 has a point be carefull of enviromental impact when doing so. You will be much happier doing it right the first time then doing it again in a year because the finish has failed.

Doesn't the surface need to be sanded to bare shiny metal in order to be primed properly? There are so many slats and edges, nooks and crannies to do. This will take forever.

Ideally yes. That is why I suggested you rent a sand blaster. As long as the prime paint that is already I assume on the metal surface and its in good shape all you need to do is scuff up the surface. The idea is to remove any rust. It is more critical on corners and edges Of course the hard parts to sand as these are the areas in which rust first.
You can purchase a small portable sand blaster at some of the big box stores that may work fine for these areas. And use a good quality sander with agressive grit to do the flat areas. But what ever area you work on be sure to remove the rust completely and prime ASAP to prevent oxidation from starting again.