I used the itch desktop app. It downloads the game and runs it in a separate window.
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I added it to Lutris and it ran perfectly from what I could tell.
Instructions (edited 2020-03-21):
1. Install Wine with your distro's package manager. Install Lutris from there too if it's available, else use the download page: https://lutris.net/downloads/
2. Start Lutris, go through the driver installation process if prompted. Click the + icon in the upper-left, Add Game. Enter a name, then for the Runner select "Wine (Runs Windows games)".
3. Click the "Game options" tab. For the Executable, Browse and select "FPA World 1.exe" from the extracted ZIP.
4. Save your added game, and try playing it!
5. Wine prompted me to install Windows versions of Mono and Gecko. I just clicked cancel on the prompts, and it never asked me again.
If the game opens with a white screen and then immediately closes, or has other odd behavior, try changing Wine to one of Lutris's builds:
1. Find Wine in the Runners pane on the left of the Lutris window. Click the left "Manage Versions" icon.
2. Select the most recent version with no addon. For me it was lutris-5.4. The build will be downloaded and extracted. Click OK.
3. Back on the Runner page, click the right Configure icon. For "Wine version" change it from your System build to one you downloaded.
4. Save, and try playing again!
There's 24 hours left in the current sale, maybe this will reach you in time!