You would probably need to aim for GBP/EUR50 per room per night for budget hotels although you can get them for as low as GBP19 but realistically you would be looking at higher than that (the cheap rooms go quickly). It is worth noting that the Republic of Ireland is not so big on the budget hotel chains (IMO) as the UK. You are more likely to be looking at BnBs which are more likely to be the EUR70 level in Ireland and since they are not chains, many of the BnBs are closed at this time of the year so it would be necessary to book ahead.
I think, for two people, you should maybe aim at GBP50/EUR60 for foos for costs for the two of you combined but that will go higher or lower depending on your tastes or budget. Most scenery, landscapes and trrekking costs are minimal or zero but do come prepared - in February last year Ireland had notable snowfall.
In terms of travel costs that will depend on how you plan to travel and to where but I would have thought GBP20/EUR30 per day or less for the two of you combined depending on whether you want to get passes. To be honest, scenery and trekkkng etc is best dealt with by car and if that is possible in terms of driving skill and budget you should go for that.
Some of the better scenery in the UK and the Republic of Ireland is less accessible by bus and is certainly more conveniently accessibly by car.