It depends on your mortgage but with nearly all mortgages the sooner you pay off more of the mortgage, the sooner you benefit from that. There is no difference to overpaying in sections or saving those up and overpaying in one lump sum.
The only difference would be if you earned interest on the savings, which was more than the rate of the mortgage (unless you are very lucky, your mortgage rate will almost certainly be higher than any rate you could get on savings especially after DIRT).
Overpaying a monthly amount with your mortgage is also a good idea in that it gets you used to the higher mortgage amount - which you will most likely have to pay when rates do rise. Also once it's paid you won't have access to it so won't be tempted to dip into it.
Saying that it's sometimes useful to have a lump sum in case of emergency so while you are being incredibly sensible in trying to reduce your negative equity and should benefit from this later when you will have more options to remortgage if rates do rise, be careful not to overstretch yourself and put everything into overpaying. An emergencuy fund could be used to pay off more of the mortgage if you need it, but in the meantime kept for other emergencies if needed.
Another thing worth considering is how much you can overpay without penalty. BOI for example have the following on their website: "Restrictions apply to Fixed Rate loans. If your mortgage is on a fixed rate the maximum monthly overpayment amount allowable is €65.00 above your standard mortgage repayment." So if you're lucky enough to be able to afford more than the maximum amount then you can still save that separately and use it to pay off in one lump sum if you ever want to remortgage, so your new mortgage would be for a lesser amount.