Of course, it's all map dependent, but in general, you want your shield up very often. The moments you drop them are when you move to another spot with cover. So keep it up when you're moving to the next corner, or behind an object. Once behind cover, drop shields to let it recover. Never let your shield break, it will take longer to recover the shield if it breaks.
The tricky part with Rein isn't the shield, it's when to charge. As a Mercy main, I have seen dozens of Rein solo charge straight into the enemy, without even looking back.
Just be patient with Rein, and be liberal with the Fire Strike. Move from cover to cover with your shield. Charge in only if you have some support with the team (e.g. Winston jumps the same time you charge in) or if you can guarantee a pick.
Good highlight by the way. Nice timing with ultimate.
edit: some heroes are good at providing you breathing room to move forward with shields down. A soldier that provides suppressing fire is great. A widow that called out picks is another queue to move forward with shields down. If key opponents are killed (turrets/bastion/symmetra) that can control specific chokes.
Playing Rein is very hard, especially in solo queue. He isn't a mechanically difficult hero, more like decision making hero. His decisions will have a huge impact on win/loss.