I tried a couple of times with West Rome until I decide to go for Eastern Rome. This is very much a defense scenario and UA for Eastern Rome is great.
First 10 turns or so were the toughest. I played them over and over again. Was it me, or was it supposed to be this hard?
Even if every turn takes a long time i don't recommend to turn off single player Quick movement/combat (except then replaying),
A good thing with replaying
(this is not "Hall of fame"...) is that you start to see how AI moves and if you play right you can use unguarded cities as bait.
I lost 6 cities during 13 turns
To the Huns
Trebizond
Theodosiopolis
(was later taken by the Sassanids)
To the Sassanids
Melitene
Edessa
To the Goths
Sirmium
Naissus
I build a lot of Dromons to keep the Vandals away
(I took Bombardment as promotions, very nice in the late game). The Vandals was only a problem in the beginning
(at least I thought...)
In Turn 21 I took back Trebizond (Horses and Gold). My Dromons was a great support.
In Turn 22: Edessa
In Turn 24: Naissus (Silver and Truffles)
In Turn 29: Sirmium (Gems)
In Turn 56: Theodosiopolis
In Turn 67, 68 and 69: Melitene
To prepare my Empire for coming policies I build markets and harbors (to be able to dig up roads and save expenses), circus and stone works. I didn't build any barracks because i was afraid of coming policies (but that was a misstake....)
The next 25 turns was two border war against the Goths and the Sassanids. After taken my cities back, the Goths never cross my border again even if he tried with large hordes. During this time I helped WR to take back two cities from the Goths.
The war against Sassaninds was more like a guerrilla war. Maybe i should have been more aggressive and taken one or two of his cities (it was hard to get my last city back)
I had one fleet of Dromons scouting on alert for Vandals
I only adopted 4 policies. I could have build barracks and still avoid Usurper General (+1 culture)...
Policy adoption:
- Barbarian Conscription
- Rise of Serfdoom
- Neglected Infrastructure
- Coin Debasement
My border against the Goths
My border against the Sassaninds. Around turn 60 the Huns come from the north, But he was easy to stop with Dromons and Spearmens. Lucky for me I was able to take Melitene on turn 69
In turn 62 Civ V warned me about enemies outside Alexandria i thought it was something wrong until i saw i Vandals, later i saw more of them. Lucky for me, i had time to move my Dromons and move two Bowman and build and upgrade a warrior. I think i scared him with my Dromons because he slowed down. Or maybe he was tired after walking through Sahara desert.
Turn 60 I have Goth and Sassaind under control and almost done with prep. for a concuer back my last city from Sassaind. But then my Dromon scout alerts me that the Huns are coming....
The 2 legions in Africa can setup a reasonably fast defense and can hold the double lux city if reinforcements arrive soon enough. The Vandals have a habit of sending a lot of stuff to Spain for one city. Never mind that whole area is defenseless with an iron supply. Their units leave Spain almost right away. 
