The thing in your link is a regular ISA sound card. It has a DA15 joystick port that also supports MIDI over some unused pins. This is by far the commonest way people used to get MIDI output out of a computer in the day. You can use a joystick-to-MIDI (DIN) cable with it. Cheapest option is to make one yourself, otherwise you need to buy one.
Note that MIDI implementations were not all equally good, and you should never expect pro musician quality from a consumer sound card. Common issues were hanging notes, incorrect notes, delays etc. The Soundblaster 16 line was particularly notorious for these problems, but a lot of other cards also suffered from them to greater or lesser degree. If you want good MIDI, you need to do some research into which sound card you choose. Personally I'd recommend something based on the C-Media CMI8330 chip, but YMMV.
Note that sound cards like this only support UART mode MIDI, not intelligent mode. For intelligent mode you would need a TSR like SoftMPU.
It is possible to find proper MIDI interfaces supporting all MPU401 features for the sort of budget you mention, but you need a lot of patience. In the retro computing/gaming community things like Roland's own cards and their MusicQuest clones are much sought-after and command a hefty price premium, but to musicians they are generally considered old useless crap and are sold at prices to match. I've picked up two MusicQuest cards within your budget that way, one for EUR 7.50, the other for EUR 15. Both have native DIN connectors you are expecting. But finding these requires a lot of patience. If you want one NOW you pay very high prices and are better off getting a replica - but including shipping that would come to 10x your budget.
But to keep it simple and get an authentic 386 experience: just buy a period-correct ISA card (avoid ISA PnP on a 386, so get something with jumpers for address, IRQ and DMA - the one you link to is a PnP card) and hook it up via the joystick-MIDI cable. Just be aware it may also be an authentically buggy experience.