I have a flight + connection on same ticket - can I book a separate first leg of the journey and still use the connecting flight booked prior?

As a rule, you can not.
Most or all airlines have the rule that if you are not on one leg of the travel, they cancel all other legs.

Your best thing will be to contact Lufthansa explaining your worries and ask them for advice. Likely they will tell you not to worry, but it might also be that they will be happy to change your ticket or even allow you to drop the first leg, but likely at a steep extra fee.
Flights with connections are often much cheaper than the same main flight without the connection and airlines do not want people to drop the connection part.

If all the bookings were made through the one provider (Lufthansa) they are responsible to get you from start to finish of the journey irrespective of any short connection time. They will book you onto an alternative connecting flight if the first flight is delayed. They have systems in place to do that. see https://www.caa.co.uk/Passengers/Resolving-travel-problems/Delays-cancellations/Your-rights/Your-rights-when-you-miss-a-connection/

Check with the airline.

I was once in a similar situation in which I’d booked Munich–Montreal–Ottawa and back. Since I was already near Montreal, I called the carrier (Air Canada, the trans-atlantic flight being a code share flight with Lufthansa) and asked if I could cancel the Ottawa–Montreal leg of my flight (or just not show up) without affecting the rest of my itinerary. They said it was no problem, except that I wouldn’t be able to get a refund on the Ottawa–Montreal leg (or use that part of my ticket later).

However, that was back in 2002. Your mileage may vary, and only the airline can tell you for sure.

The OP's journey is history now, and I trust the connection was made. However my policy is to refuse tight connections -- I always demand at least two hours between flight legs and three hours if one of the legs is intercontinental. With ever-increasing security checks between legs of flights an hour's delay at security is certainly not unheard of, and if flying into the US from an airport which does not have immigration clearance abroad, the immigration delay can be well over an hour.

You're mis-reading your booking itinerary. Your itinerary will show both flights on the page, but you'll be issued the needed boarding passes for your flights at the airport. The itinerary that you were provided is only just that. Isn't a "Ticket". When you get to the airport, you'll check in kinda like a hotel, and you'll then be issued boarding passes while checking your luggage. Also, 1 hour is more than enough UNLESS your flight from Dublin is late. I myself have made connections in much bigger and busier airports than Munich in about 1 hour. So don't fret my friend.