Rise and arise are close synonyms, although arise has another meaning that you have already given. They can both be used to mean "get up", and have very similar etymologies as can be seen below.
Older translations of the Bible certainly use arise as an injunction, as can be seen in Bibile's quotation from Ephesians 5 (English Standard Version):
“Awake, O sleeper,
and arise from the dead,
and Christ will shine on you.”
This is very much in the poetic register, and the translator makes full use of the alliteration that the usage provides. There is also, perhaps, an allusion to both meanings of arise, i.e. to "get up" from the dead, and also to "come into existence" from the dead.
Helpfully, Bibile also quotes from Isaiah 60, where both meanings of arise are also employed:
Arise, shine, for your light has come,
and the glory of the Lord has risen upon you.
For behold, darkness shall cover the earth,
and thick darkness the peoples;
but the Lord will arise upon you,
and his glory will be seen upon you.
And nations shall come to your light,
and kings to the brightness of your rising.
It may be useful, when reading older translations of the Bible, to look for both meanings whenever you come across the usage of arise.
Finally, some useful comments are made in Biblical Hermeneutics.SE on Isaiah 60:1, where a literal translation from the Hebrew is given as "Rise and shine!", a (still current) English idiom meaning "Wake up, and get out of bed."
Online Etymology Dictionary: rise
Online Etymology Dictionary: arise