Erst die Rolle klären, dann ARIA setzen

<button type="button" id="sort-button"
  aria-haspopup="menu" aria-expanded="false"
  aria-controls="sort-menu">
  Sortieren
</button>

<div id="sort-menu" role="menu"
  aria-labelledby="sort-button" hidden>
  <button type="button" role="menuitem"
    data-sort="name" tabindex="-1">
    Nach Name
  </button>
  <button type="button" role="menuitem"
    data-sort="price" tabindex="-1">
    Preis aufsteigend
  </button>
</div>

<ul id="courses">
  <li data-name="Töpfern" data-price="55">
    Töpfern · 55 €
  </li>
  <li data-name="Holzschnitt" data-price="40">
    Holzschnitt · 40 €
  </li>
  <li data-name="Fotolabor" data-price="48">
    Fotolabor · 48 €
  </li>
</ul>

const trigger = document.querySelector('#sort-button');
const menu = document.querySelector('#sort-menu');
const items = [...menu.querySelectorAll('[role="menuitem"]')];
const courses = document.querySelector('#courses');

function openMenu(position = 0) {
  menu.hidden = false;
  trigger.setAttribute('aria-expanded', 'true');
  items[position].focus();
}

function closeMenu(restoreFocus = true) {
  menu.hidden = true;
  trigger.setAttribute('aria-expanded', 'false');
  if (restoreFocus) trigger.focus();
}

function sortCourses(key) {
  const rows = [...courses.children];
  rows.sort((a, b) => {
    if (key === 'price') {
      return Number(a.dataset.price) - Number(b.dataset.price);
    }
    return a.dataset.name.localeCompare(b.dataset.name, 'de');
  });
  rows.forEach((row) => courses.append(row));
}

trigger.addEventListener('click', () => {
  menu.hidden ? openMenu() : closeMenu();
});

trigger.addEventListener('keydown', (event) => {
  if (event.key === 'ArrowDown') {
    event.preventDefault();
    openMenu();
  }
  if (event.key === 'ArrowUp') {
    event.preventDefault();
    openMenu(items.length - 1);
  }
});

menu.addEventListener('keydown', (event) => {
  const current = items.indexOf(document.activeElement);
  const last = items.length - 1;
  const destinations = {
    ArrowDown: (current + 1) % items.length,
    ArrowUp: (current - 1 + items.length) % items.length,
    Home: 0,
    End: last,
  };

  if (event.key in destinations) {
    event.preventDefault();
    items[destinations[event.key]].focus();
  }
  if (event.key === 'Escape') {
    event.preventDefault();
    closeMenu();
  }
});

menu.addEventListener('focusout', () => {
  requestAnimationFrame(() => {
    if (!menu.contains(document.activeElement)) closeMenu(false);
  });
});

items.forEach((item) => {
  item.addEventListener('click', () => {
    sortCourses(item.dataset.sort);
    closeMenu();
  });
});