Can I use "and" many times in one sentence?

You can certainly use more than one and in a sentence, and the rhetorical device is called polysyndeton.

At school today, we had sums and writing and play and dinner and a story and a prayer and then I came home.

Good writing style usually advocates a variety of sentence lengths. Some can be very short. You can certainly use and to join two related clauses together, and within one or both of those, you might use and as well.

One way of expressing what you want is as follows. I've also indicated changes in bold. Separate the different ideas into discrete sentences. If necessary, separate each of those sentences again.

I appreciate that all my previous work experience is within office based industries but I am a quick learner, keen and enthusiastic, so although the change appears drastic, I am confident that I can manage it. I am looking to change to ensure longevity of career and also that the remainder of my working life is spent doing something that I enjoy and can feel passionate about.

I appreciate that all my previous work experience is within office based industries but I am a quick learner, keen and enthusiastic, so although the change appears drastic, I am confident that I can manage it. I am looking to change to ensure longevity of career and also that the remainder of my working life is spent doing something that I enjoy and that I can feel passionate about.

You'll need something more concrete than "I am confident that I can manage it", of course.

There is nothing that would stop you from using 'and' multiple times, but if you are to list things, using a comma is better. Example: "I like to play football, ride my bike, play guitar and hang with friends." Instead of "I like to play football and ride my bike and play guitar and hang with friends".

In your case, you are listing qualities for what I assume is a resume for a job. In this case you want a clean text, using 'and' then like you've done doesn't really help that out, I'm afraid. Try instead to list your qualities as demonstrated in my example, put your finest quality last. Such as: "I am a quick learner, keen, enthusiastic and looking to change in order to ensure longevity of career, to ensure that I will keep working with what I enjoy and feel passionate about."

To kinda sum up my point, there is no actual rules saying you can't use 'and' too much, but it is certainly a wise idea to keep the use of it down so your text looks cleaner.

If you had a sign writer write a sign for your pub and you got this: The fox and the hen. You might say to the signwriter: "You have left too much space between fox and "and" and "and" and the hen".
That is and 5 times and consecutive.