The main challenge of facing a mummy is their powerful offensive attack. Every few seconds, a mummy will shoot a swarm of flies out of its mouth, which scatter like pellets from a shotgun, but much slower and wider. Each fly deals 2 damage and there are 8+ in a swarm, which are fired in a broad enough pattern that the swarm is typically impossible to jump over. Being caught full-on in this attack is almost always fatal. Note that the flies can be blocked by holding a dead enemy and facing the fly swarm.
Luckily, mummies have limited mobility: they are fairly slow-moving and cannot jump. In addition, their attack can only fire directly forward, and the flies they fire disintegrate when they hit an obstacle. Therefore, the best strategy for dealing with them is to get to a place where their attack cannot reach you, and plot your offense.
The most common scenario is that you have found the mummy, but there is some obstacle between you and it. This is where it is useful to exploit the cooldown time on the mummy's attack. When a mummy has acquired you as a target (ie. you are in its line of sight) it will pause in its patrolling to spew its fly swarm. If you are still in sight after the initial attack, it will pace towards you for 4 tiles and attack with the flies once more. This break is your window of opportunity to attack. Specifically, you must step out from your cover to draw the mummy's attack, use your cover to avoid the attack, and then move out to counterattack.
This might be as simple as jumping out to shoot the mummy with your shotgun or pistol before returning to cover, repeating as necessary. Alternately, a well-placed bomb will instantly destroy the mummy. Given the monster's slow, generally predictable movement, this can be a surprisingly simple way to defeat it. Sticky bombs thrown directly at the mummy can make this even easier. Thrown skulls are also effective. If you are desperate you might instead attempt a jumping chain; try to bounce towards the back of his head and do your darndest to avoid the fly swarms. In fact, once mastered, this technique is extremely safe assuming you can get to the mummy fast enough.
The other, safer scenario is when you are above the mummy and there is an opening leading to it. This is your opportunity to get clever with bombs; with good timing, and knowledge of the trajectories of thrown bombs, you can explode a mummy without ever exposing yourself to its line of fire
The one redeeming feature of Mummies is that they can drop the scepter, which is one of the best weapons in the game. However, the scepter will only drop from the Mummy on level 13 of the regular game; mummies in custom levels or later levels of the Temple will give you no reward.
Once you've obtained the scepter, killing further mummies is a snap, as the scepter will kill them instantly - along with just about everything else. Note that if you are doing a City of Gold run, you must obtain the scepter on level 13 as the final step to the City of Gold, to open the entrance to it on level 14. However, this will use up the scepter, leaving you weaponless for the last two floors, unless you find a different one.
If the Mummy sinks into a lava pit, the Scepter can still be recovered by draining the pit with bombs.