I have been looking at the Akathist link and then pondered your question to Father Deacon and I realized that most of my daily prayers are in this format where it needs to adjusted for the layperson and I say them without thinking about it. I also realized that I do not know of any written adjustment for the layperson is in print. There are Typikas in print for when a priest is not present at Sunday Divine Liturgy but that is rather more important.
As I look at your Link though I thought I could tell you a bit how I go about saying such an Akathist. I won't go through the whole thing because it will get redundent.
At the begining the top line is said and insted of the priest saying "Blessed is Our God..."I start with the sign of the cross with the words
"Through the prayers of our Holy Fathers, O or Jesus Christ, Our God, have mercy on us."
Unless it's During the Easter seaon and we sing three time "Christ is Risen From the dead, by death He conquered death and tho those in the graves He granted life."
Say everything else until you get down to the First Chant. Say all the lines but without the response.
At the Kontankions, say the prayers but not the responses.
This contnues down through all the Chants untill the dismissal. The only thing not said is "Glory to you O Christ..." I start the dismissal with the "Glory be to the Father.+.." but I don't say "Give the Blessing".
The end prayer is where I am the most unsure. I say the prayer which is the same dismissal for the office of the hours inwhich case I know who the saint of the day is. if I don't know I just say "...Illustrious Apostles and through the prayers of all the saints for He is gracious and he loves mankind!"
One more sign of the cross and a silent Amen and I am done.
I hope that helps. Remember it is prayer and it is really hard to mess it up.
In the Byzantine Catholic tradition think about getting some icons for your home. Place then on an eastern wall. This is the Icon corner of the Byzantine home and a place where we can go and pray in peace with all the saints and our Lord Jesus Christ. This is a good link
for information about Icons.
Hope this helps a little bit.