I have a LH ticket to fly from Sofia (non-Schengen) to Luxembourg via Munich, and my transfer time is 1h 10 min. Would that be enough time to be able to pass border and reach the departure gate? Am I going to have to switch terminals?
LH uses a terminal in Munich that has been built to their specification to make such short transfers possible. 1h 10min is actually rather generous even, as the minimum for Munich is 30 minutes.
So yes, you will be fine. And in the unlikely event that your inbound is delayed LH will put you on the next available flight.
Under standard conditions, the transfer time is already plenty. Under current conditions, it should be even more than enough.
All Star Alliance flights (that includes all Lufthansa flights; excluding flights to Israel) arrive and depart from terminal 2. In essence, terminal 2 is a long building with a passport checking area in the middle. A satellite terminal also exists, which is connected to the main terminal by an underground people mover also around the centre of the main terminal. The satellite being smaller, the walking distance from the furthest gate of the satellite should approximately equal that from the most extreme gate in the main terminal.
The terminal is separated into an upper level (non-Schengen departures and (some) arrivals) and a lower level (Schengen departures and arrivals). You can transfer from one level to the other at the centre of the main terminal where immigration is performed. The gate numbers are the same on both levels but differ by their leading letter (Schengen: G; non-Schengen: H; Schengen satellite: K; non-Schengen satellite: L). If you're concerned, check what gate numbers you pass on your way to immigration, compare that to the gate number your connecting flight has, and you will have an immediate idea whether to go left or right afterwards, saving you another minute or two.
When arriving, make your way to the centre part following the signs and pass through immigration. You will then need to go down from the non-Schengen storey to the Schengen storey, locate your gate and make your way there. There should be no problem making it in about half an hour unless immigration is really congested. (Assuming you are an EU citizen, you should be able to pass through quickly anyway.)
There have been reports of people being rushed from gate to gate with help of Lufthansa employees in the event of a delayed inbound flight to ensure the connection is made.
Don't hesitate to alert the flight crew of your first flight that you have a connection in case you are concerned.