Div with vertical and horizontal text inside

You almost had it. Added flex-direction: column and switch your align content and justify content on your container div.

.container {
  width: 70px;
  height: 400px;
  background-color: lightgray;
  display: flex;
  flex-wrap: wrap;
  flex-direction: column;
  align-content: center;
  justify-content: space-between;
}

.circle {
  
}

.bottom-container {
  display: flex;
  flex-direction: column;
}

.artist {
  border: 1px solid red;
  writing-mode: vertical-rl;
  transform: rotate(180deg);
}

.number {
  border: 1px solid lime;
}
<div class="container">
  <div class="circle">🔴</div>
  <div class="bottom-container">
    <div class="artist">Artist - Never forget my love</div>
    <div class="number">03</div>
  </div>
</div>

The circle div should take full space. Adding width and text-align to the circle class did the job:

.circle {
  width: 100%;
  text-align: center;
}