Monica - what you appear not to be doing is testing the fuse you take out each time - put it in a table lamp plug and see if the fuse is still good - if it is you can stop worrying about fuses and concentrate on the real fault - you will also save a few pennies on fuses.
Your 500W mower will operate fine on a 5A fuse - it's nothing whatsoever to do with personal safety from electrocution - a fuse is there to stop the wiring getting too hot and catching fire - it takes only a few milliamps to electrocute somebody.
My suspicion is that your mower - if it cuts out on a 5A fuse - has an electrical connectivity problem - a bad connection somewhere - in the plug, the cable, the control switch or in the internal wiring.
If it does not cut out on 5A then problem solved - when motors start they usually have an inrush current somewhat higher than normal operating current - a 3A fuse may well pop.
And if for some technical reason you are forced to use a 13A fuse you will be quite safe.
The key safety thing when using electrical equipment particularly outdoors is to have an earth leakage breaker (RCD) somewhere in the circuit - these do prevent accidental electrocution.
Good luck...
Peccavi
April 2010