(1) Quit the job and do other business
IMHO you should not start a business if the only reason for doing so is to run away from your current job. However if there is a compelling business case for having your own start up, then .. yeah .. why not.
(2) Face the challenge :
Yeah. Face the challenge and enjoy doing it.
Quit the job and do other business
Never do this. If you are confident in taking up these challenges these will not only boost your confidence but also will prove your talent in the company.
If you are not confident in finishing these tasks in time then talk to your management persons about the tight schedules and dead line and bring a suitable plan in which you can complete these tasks without any issues.
I was put in a not dissimilar situation with my first job and in many ways it was the making of me. If you're any good at this game you'll rise above the challenges and it can push you years ahead in your career. So option #2 for sure.
The catch is that you are inexperienced and being asked to do a great deal. Your employer should have no expectation for you to perform at this level, and the risk is entirely theirs. I would expect most people put in your position to sink, not swim, and it therefore follows that so will the company. If you have anyone senior to still follow and get advice from then do so. Leverage everything you can, because if you spend most of your time with clients then you aren't spending it learning to code.
And remember that because the expectation is unrealistic, if you do succeed here then you damn well should be rewarded for such when the company starts making profits again.
The company you work for has given you quite a gift. They had a bunch of experienced people, and decided to fire them to save money, not realising that a company has to actually make money. Extremely dumb of them, but no reason for you to complain.
This is all good for you - you just got promoted. Okay, so the money you get probably hasn't improved, but you will now get rapid experience across every aspect of your career domain.
Think of it as the best training you'll ever receive, and someone else is paying you to do it!
Face the challenge, it's the best way of learning, under pressure.
Learning to cope with this pressure will be a good insight into this industry and if you don't like it you can get out early doors without wasting ten years or so then deciding this industry not for you!
good luck