This achievement is highly skill-based so there's not much I can offer in terms of a step-by-step solution. I'll instead give some tips and explain what worked for me.
Apocalypse mode is brutally difficult; you go much faster and hazards are more dangerous and plentiful. And the between regions bird will often drop a stone cube. Bastard.
I saved the Apocalypse achievements until after I had completed everything else in the standard game and my plan was to dedicate a few hours one day to studying the daily Apocalypse layout in order to develop a path through the regions that would get me past region 6. Here is my successful run:
For upgrades, I went with the magnet, tighter turning, and extra jump storage. Extra escape portal storage may be worth considering if you can find a path that allows you to collect more than one.
This particular daily layout allowed me make use of early region portals in regions 1 and 2 so I could basically skip straight to region 3. And my region 3 ended up having an amazing, easy to follow path through several jumps and boosts. It didn't take long for me to get this portion of the run down perfectly.
Things started getting dicey in region 4. Through exploration, I found that pushing hard left as soon as region 3 ended (ignoring the between region bird entirely) put me at an escape portal pickup. My path from this point forward was mostly random and it took me a bunch of runs to figure out how to best approach the hazards in the upcoming regions.
Just like with the standard game, the regions are generated each day with similar hazards so only through repetition will you learn how to best navigate through/past/around them. My region 4 consisted of a loop through the two mountain tight squeeze, a brief area with pillars and explosions, and a path through falling pillars that were avoided by going side/center/side. After some practice I was occasionally getting to region 5 and after more practice I was able to consistently get there without losing my escape portal.
I needed to hit some boosts in region 5 to get the sun high enough that I could try to find the border path between areas in region 6 and coast through to the end. An (un?)lucky reaction to a moving hazard ended up launching my very high into the air, which probably got me safely past a decent chunk of the region. It felt like it took me forever to get back to ground level, though. I ended up starting region 6 with the bottom of the sun just sinking below the horizon, an escape portal, and no jumps.
I got a boost and a jump early in region 6 that I immediately had to use, but I ended up crashing right after and using my escape portal. Once airborne, I pushed hard right, went through a boost, and miraculously ended up in the border area that I was hoping to find. This area is a somewhat safe path between sections of hazards and offers a mostly direct run forward but contains no tris or powerups. As the sun sunk further below the horizon, I hoped that it would afford me enough energy to get through the region, and it did. Barely. I ran out of steam during the between region area but the achievement popped as soon as region 6 ended.
If you plan on working on this achievement, I don't think trying for, say, 30 minutes daily is the way to go. Instead, plan on dedicating a few hours to working out a path through the daily layout and if you don't make some decent progress, try another day.
Good luck. You'll need it.