Glasnevin Cemetery is less than a 10-minute walk from Eganshouse. Everyone and anyone from Irish History has been buried here. I would highly recommend the historical tour of Glasnevin cemetery. The ticket includes a tour of the cemetery, stopping at various tombstones of interest and free entrance to the museum itself.
You can then take a stroll around the National Botanic Gardens which you can access directly by foot from the cemetery. There are cafes in both places so you will be spoiled for choice. If you are feeling more adventurous you can catch the red hop-on, hop-off bus from the cemetery and this will bring you on a guided tour of the surrounding area and city centre. The bus will bring you back to the cemetery when you finish. If you are staying at Eganshouse this will be a bonus for you!
Enjoy your trip!