HI,
I am/was in the same boat as you so I started going to a lot of meetup and eventbrite meetings. Nearly every one I went to I found something new. This could be a starting point for you. As an example I went to a WordPress meetup in my local suburb, there are only about 10 people who go but 2 of the guys that present each month not only have WordPress but wealth of information on most aspects of the web world. They bend over backwards to help an any aspect or they will know where to get information.You would pay hundreds of dollars for a consultation out side the group and I get it for free. How good is that :)
Regards

Peter
 

Good advice above about meetups, I run a monthly event which any one can attend for free -

http://www.meetup.com/Online-Marketing-Sydney/ 

But to Answer your original question,

What computer knowledge do you need to run a start up:

- It really depends what your start up is? What is your budget?

Is it mobile startup, offline, online.

To be hones you need to know the basics like email, excel, power point, MYOB ect ect to deal with people.

Most things you can pick up along the way, if you have funds you can outsource things like SEO, SEM, Web Design, Hosting.

I deal with start ups where the founders are not "Computer Savvy" they are more "Business Savvy"

Peter and James have covered it quite well. The easiest way to learn is to learn :) Talking to people that can explain it to you simply is probably the easiest way to learn. And everyone is more then happy to help out if you ever have a question :)

The most important skill for you to have as a startup owners is brushing up on your ability to find the right people to outsource it to. Knowing a general gist to help you find out if you've paid the right person to do it is probably the best and most time saving solution.

That being said, here is my nerdy answer:

I personally learnt it through building my own site, but that might be overkill. If you really want to learn it thoroughly in practice sometimes starting a mini website project on the side may teach you a wealth of information. Make sure the site is complicated enough to cover all the things you want to learn. You'd be surprised how fun it is to learn as you are doing it.

I learn faster when I do it and run into a problem, then I scour Google and once you apply it, it makes a whole lot of more sense then just reading about it.

For this purpose I can suggest you start off by going through the installation process of: 

• Wordpress - fairly easy to install and you can enjoy a great blogging platform on the other end
• Magento/OpenCart/Other open source ecommerce engines - These are fun if you are looking to get into ecommerce.
• Custom Site - This is more for advanced learning you can pick a language and learn from scratch. There are plentiful and abundance of tutorials online. I also suggest somewhere like codeschool which teaches you basic code through easy and fun modules and quizzes online. Some basic languages which are popular to learn are: HTML/CSS (basic), PHP(a very popular language with more features), Ruby on Rails (A fast deployment language that twitter used to start).

There are also a bunch of different languages that are more to do with application development (both software and phone and computer apps)

Of course learning code can be more learning than you need, but knowing what exists and having a general idea will help you when you are instructing someone to build a website for you :)

First, read about the technical aspects of any startup, Google will help you find answers. For example, if you want to make a regular site, you can help site designers - Tilda, Vicks, figma. 
You can also take courses like htmlacademy where you get the opportunity to see the typical tasks that occur.
Another good opportunity is communication with specialists. Do not underestimate the importance of Analytics from experts. Now we are talking about digital agencies. Their contribution at the start of your project can play a key role and significantly increase the conversion of your startup.