iOS 9.3.5 in iPad Mini 1 I have a first-generation iPad mini (model MD529LL/A, A1432, 32GB) running iOS 8.4.1. I use it mostly for web browsing with Safari and Chrome, email with Gmail, reading PDFs with iBooks and Adobe Acrobat, maps with Google Maps. It's not the fastest experience but it works just fine. 

However, some apps no longer support iOS 8 and I'm considering upgrading the iPad to iOS 9.3.5, the latest version supported for this model.

Will this upgrade make my iPad feel much slower?

What other downsides should I be aware of?
I think you can expect some slowdowns, especially in the time it takes to launch some apps. Here's an excerpt from an article that compared iOS 8 and 9 on the iPad mini:

The decrease in app responsiveness is a disappointment considering Apple's claims, but we have to consider that we are dealing with old hardware that just doesn't seem to handle Apple's new Metal API/compiler particularly well. At least flicking through home screens is reasonably smooth, even if using complicated apps isn't - scrolling through the Guardian app on the iPhone 4s or iPad mini is surprisingly juddery.
At least the iPhone 4s gained some battery life from the update to iOS 9 - the 40 minute (approximately 9 per cent) increase in our Web use test, as well as Low Power Mode, suggests that in everyday use you'll be less likely to feel battery anxiety. The iPad mini showed no signs of extra life after updating to iOS9. We should stress that both devices on test have been in constant use since their launches three and four years ago, so both devices have suffered battery fatigue. We're only interested in the comparative times, allowing for a few per cent margin of error due to climatic conditions affecting the battery chemistry in the devices.

Something further to consider is that not all new iOS 9 features are enable on the 1st gen iPad mini. Again, from the same article:

Apple's latest iOS brings plenty of new and useful features, but most are confined to newer hardware. For example, none of the new multitasking features are available for the iPad mini. You'll need at least an iPad mini 2 or iPad Air to invoke picture-in-picture and Slide Over, while only the iPad mini 4, iPad Air 2 or iPad Pro can handle side-by-side Split View.

So, you'll have to balance that against the increased security, new features that are enabled, and increased app compatibility that iOS 9 will bring to your iPad mini.
And, if you end up installing iOS 9 on your iPad mini and regretting it, downgrading back to iOS 8 is no longer an option. So, one other downside (if you can really call it that) to consider is that iOS 8 to iOS 9 on your iPad mini will be a one-way trip.