If Venus were ejected past Pluto, it seems that within a relatively short time, it's atmosphere would be CO2 ice- a mile or two thick.

Give Venus a moon and perhaps that relatively short time might be millions of years or the atmosphere doesn't freeze out completely. It depends upon what kind moon you are giving Venus.

Earth's early moon- which according to one theory formed close to Earth- would probably keep a planet fairly warm.
And btw, could be part of the solution to why Earth was warmer with the younger sun.

Without sunlight one needs something which is providing energy.
We know of only a few types of energy sources:
Fusion from the Sun. Fission from radioactive decay. Gravity such as in terms of tidal or orbital energy. And chemical, such as something oxidizing [or any exothermic reaction].

Having a planet far away from a sun removes the dominant energy source which we are familiar with on Earth- the fusion energy from the Sun.
[Btw, if we lived a mile below the surface, the Sun's energy would not seem as dominant. At first glance the Sun energy would seem almost irrelevant or insignificant.
But if you removed Earth from the effects of the Sun's energy, the average temperature at a mile below the surface would be significantly affected over significant periods of time.]

Let's return to Venus without a Moon.
If ejected from the solar system, how long in more exact terms: days, weeks, months, etc does it take before say, Venus' atmosphere becomes mostly liquid?

Currently Venus is emitting a huge amount of energy- roughly the same as it's receiving from the Sun.
As Venus goes out of the solar system, at say earth distance, it will receive about as much energy as the Earth but be emitting far more energy than Earth does [Venus will be cooling].

How long would it take for Venus to emit the same amount of energy as Earth does and what is happening on Venus when it reaches the point where it's emitting the same level of energy as Earth?

Without sunlight, an 1 atm atmosphere does hold heat in any significant manner- though Venus' atmosphere is more like a ocean than an atmosphere**. In terms of mass it's +90 atmospheres is roughly like +2970' deep liquid ocean.
But in terms of latent heat comparisons of CO2 and water,
water has about 4 times the energy.
So compared to Earth's ocean, Venus' ocean-like atmosphere has about 1/4 of the energy.

So, roughly if Venus temperature is the same as earth it will cool 4 times faster. The fact that Venus is much hotter mean it cools at an even faster rate.
And it doesn't stop cooling until the total energy output equal the input.
Anyhow, one needs to do the math or perhaps someone has already done it to show about how long it would take before Venus is an ice ball.

Considering also that Venus rocky surface is also very warm and massive, it's interesting puzzle of which planet: Earth with it's moon or Venus without a moon would have it's oceans mostly freeze solid in the shortest time period and how many years would this take.

Though it seems to me that in terms of millions of years Earth with it's Moon would remain significant warmer than Venus if one measuring a mile or so below the rocky surface.