You should have absolutely no problems taking insulin and your needles through airport security.
I have taken my pens onto hundreds of flights and only once did security make any comment. This was some years ago when the pens were metal. As they are now plastic, they do not appear on x-rays so airport security have no interest in them. Likewise m they have no interest in any needles or blades shorter than 6cm.
Some people tell security and the airline than they are travelling with diabetes supplies. This may entitle you to an extra item of hand luggage if you want it. I don’t tell anyone my private health issues unless explicitly asked.
If you have reusable insulin pens (which I would highly recommend), you may want to put your spare insulin cartridges in a clear plastic bag with your toiletries. Sometimes, airport security will pull out my insulin if I do not do this.
As with all travel and trips away from home, make sure you have at least twice as much of everything than you think you need. This mean extra insulin, test strips and needles as well as spare pens (assuming you have reusable pens) and batteries for your meter if it uses them. This should all be in your hand luggage as hold luggage can go awry.
I suggest taking something for sharps. This does not need to be an official sharps bin. I tend to take a small bottle and decant my used needles into my sharps bin when I get home.
My other advice would be to ensure your hypo treatment is with you in your seat not the overhead locker. Experiencing a hypo whilst the seatbelt sign is on and realise you need to get up for your hypo treatment is more than frowned upon by aircrew. I know this from experience- a mistake I only made once.
Finally, assuming you are going to Budapest for pleasure, have fun!
