You will need an island hood they are more expensive that a wall mounted one and you need enough CFM to exhaust your grease, jeat and moisture when cooking. Usually your cooktop instuctions also give you min. requirements for hood. If you can vent straight up through the ceiling you can probably go with the min. but the minute the venting has to go a great distance or make turns you will need to increase the CFM. I always ask is there no outside wall to place that cooktop they really are not great in an island

so if I get an island hood, what size is the most ideal? 36" or 42" the vent is going straight up through the ceiling. How do I measure noise level?

The people that are selling you the cooktop should have the requirements for the hood.

You don't have to measure the noise level, You can't talk over them.

And depending on the CFM, you may have to have an air intake to replace the air removed.

Best to put it on an exterior wall.

Because an island hood isn't assisted by wall and cabinets - they help direct air flow - it's best to go 6" wider than your cook top so you should opt for a 42" wide hood. Island hoods are generally deeper than perimeter hoods for the same reason. They need to cover a larger space for better capture.

Before we chose to change up our lay-out and move the cook top from the island to the perimeter (best decision ever!), I researched island hoods. As stated above, they are expensive. Thankfully you can go straight up through the ceiling and the roof so that helps minimize install costs.

I did a mock up with cardboard and hung it from the ceiling at the maximum recommended height above the counter and lived with it for several weeks. It felt like there was a big ol' thing hanging over my head. Which there was. =) It definitely affected sight lines in my kitchen and it cemented the decision to move the cook top off the island.

Definitely do a mock-up. I'm not trying to talk you out of it but it's always a good idea to be as informed as possible for how things will look and feel post-remodel so that you end up with a kitchen you love and very few regrets.

IME, all hoods are rated for decibels and you can find comparisons of noise levels on line (comparable to a vacuum cleaner, comparable to normal talking, etc). The downside is that there isn't a standard way to measure so it's not always easy to compare apples to apples. But since you're going up through the roof, you can minimize noise a lot by going with a remote blower. That moves the fan noise to the roof.

Definitely check into MUA (Make Up Air) requirements and costs and how it works in your area. If you're in a cold weather area, I think you're required to hook it up to your furnace so that it draws in warmed air not cold air directly from outside. That adds to the cost.

4 by 8 seems a bit small for an island of this type.