I don't have a server, so i don't have crontab access. I am thinking of using a PHP script to check current time and how much time has passed. For example, I run the script, it stores the current date in MySQL and check if 30 days have passed. If so, do my stuff.
Is it possible to do all these without MySQL? And of course it is only my thinking, i haven't tried yet.
Keeping script running:
The issue is that you've either got to keep that script running for a long, long time (which PHP doesn't like) or you'll have to manually run that script every day or whatever. One thing you could do is write a script to run on your local machine that accesses that PHP script (e.g. using the commandline tool 'wget') every minute or hour or whatever.
If you want to have a long-running script, you'll need to use: http://php.net/manual/en/function.set-time-limit.php. That'll let you execute a script for much longer.
As noted in another answer, there are also services like this: https://stackoverflow.com/questions/163476/free-alternative-to-webcron
Need for MySQL?
As for whether you need MySQL - definitely not, though it isn't a bad option if you have it available. You can use a file if required (http://php.net/manual/en/function.fopen.php) or even SQLite (http://php.net/manual/en/book.sqlite.php) which is a file-based SQL database.
As i understand, you can only run php scripts, which involved by user request.
I tried this once, but it is dangerous. If the scheduled process took too long time, user may be interrupting it, and hanging up the script, causing half-processed data. I'm not suggesting it for production environment.
You'd need some kind of persistent storage, but a simple file should do the trick. Give it a go and see how you get on. :) Come back for help if you get stuck. Here are some pointers to get you started:
Take a look at Drupals Poormanscron module: http://drupal.org/project/poormanscron. From the introduction text:
The module inserts a small amount of JavaScript on each page of your site that when a certain amount of time has passed since the last cron run, calls an AJAX request to run the cron tasks.
You can implement something like this yourself, possibly using their code as a starting point. However, this implementation depends on regular visits to the pages of your website. If nobody visits your website, the cronjobs do not get executed.
You could use a webcron (basically a cronjob on another server that calls your script on a given time) https://stackoverflow.com/questions/163476/free-alternative-to-webcron