It seems you're asking two questions.
Do we need to apply the cumulative update
Yes. It clearly states (on the link provide by Atila) that you can upgrade from SDL Web 8 "with Cumulative Update 1 applied". I'd suggest it probably makes sense to apply the cumulative update anyway - it's likely you're simply working around a few problems (or even unaware of them) currently.
{do} we need to do fresh installation of Sites 9 in new servers matching prerequisites
You don't have to but if you have the opportunity it's likely much cleaner.
- Trying to upgrade your infrastructure in-situ will present a number of possible complications (if technically feasible - depending on your old/to/new versions); will you break 8.1, how long can you be offline
- If you have the opportunity then you could stand-up a new environment and bring across the upgraded database etc., here you have a plethora of options but you'll be working in two environments trying to keep content aligned along with publishing etc. (beware the workflow too)
In this scenario do we need to still go with 8.1.1
Yes. If you're wanting to upgrade from 8, then per the documentation you need the hotfix.
Depending on your timelines, budget and risk adversity, I would certainly consider upgrading to 8.5 and taking advantage of the rolling upgrades. Of course, it depends on any many factors. If you've a lone CM with a simple CD you've a different set of considerations/constraints than an organisation with scaled out CMs and scaled out CDs with a mass of GUI/other extensions and Workflow and Translation in the mix.