Suppose we wanted to do a very small experiment in (or very close to) the Cosmic Microwave Background Radiation rest frame, which we are travelling at 368 ± 2 km/s relative to. Without considering the complexity of measuring and transmitting the results back home, how would we get a 1 gram payload to such a high delta-v in the first place? What would it cost?
Is there a way to naively extrapolate costs based on required delta-v?
With current technologies, this is unfortunately well outside our reach. However, there is promise on the horizon!
Chemical Engines
The Tsiolkovsky equation is always your friend when calculating Δv for conventional engines (or your enemy, depending how you look at it!):
$$ \Delta v = I_{sp} \times g \times \ln \frac {Mass_{full}} {Mass_{dry}} $$
Rearranging to solve for fuel ratio gives us:
$$ Ratio = \frac {Mass_{full}} {Mass_{dry}} = e^{\frac{\Delta v}{I_{sp} \times g}} $$
It's that exponential that causes us problems. Even if we use one of the most efficient chemical engines in history, the Space Shuttle Main Engine ($I_{sp}$ ~ 452s), ignore its mass and ignore the mass of all the tanks/plumbing/other structure, we get a lower-bound of $Mass_{full}\approx10^{33}$kg or 1000 times the mass of the Sun. When we include all the required structure to hold all this fuel, this gets even worse!
We could cut this number significantly by making use of staging, but it's clearly not going to give us anything possible, let alone affordable. So we have to go for higher efficiency.
High-efficiency engines
If we use one of the highest efficiency engines flown, Dawn's ion thruster ($I_{sp}$ ~ 3100s), and include the mass of the engine and tanks (8.2kg engine, tanks based on square-cube from 450kg fuel : 19kg tanks), we get $Mass_{full}\approx 5\times10^{14}$kg - still totally unfeasible.
But we can do better.
ESA's in-development Dual-stage Gridded Ion Thruster (DS4G) has been calculated as achieving an $I_{sp}$ of around 20000s.
Swapping Dawn's ion propulsion ion engine for one of the same mass with an $I_{sp}$ of 20000s will get us an enormous 82km/s! If we add more fuel and scale the tank's mass accordingly, we can achieve our 368km/s with a total craft mass of ~6000kg - totally achievable!
Dawn cost around \$450m, so I'd speculate a very rough conservative cost of \$1b for building and launching our hypothetical craft. Economy of scale saves us money on the larger mass and the launch costs won't be significantly more. This obviously ignores any costs from developing the dual-stage technology which would be very difficult to estimate.
Other technologies
We can see that whatever we try, the rocket equation is always going to bite us at some point, so why don't we try something that doesn't require propellant?
Breakthrough Starshot is a proof-of-concept technology that can supposedly achieve speed far in excess of our 368km/s - on the order of 0.1c! It uses Gigawatt (read: peak power draw comparable to large countries) ground-based lasers to propel tiny crafts with extremely high acceleration.
This kind of propulsion would be ideal for your proposal - the craft would reach the required speed in a very short time, minimising corrections needed for gravitational influences and negating the need for large transmission systems.
The kind of infrastructure infrastructure would clearly be incredibly expensive - probably on the magnitude of the infrastructure budget of whole countries - \$100b - \$1t.
However, Breakthrough Starshot is relying on the costs of components dropping significantly and efficiencies increasing as the technologies progress. Some estimates give a single mission cost of $5-10b in 2036, with speculative drops in cost. Again, this doesn't account for the cost of research and development.
Note - I've tried to make some speculation and estimates on the costs involved, but they should all be taken with a pinch of salt. Also, since the 1g payload is unspecified, I'm assuming it can be modified to suit the requirements of the craft
Jack did a great job describing how to do it using propulsive engines. I have a different answer:
We already (almost) planned to do it (inadvertently).
The original plan for the recently launched Parker Solar Probe was to do a gravity assist at Jupiter for a subsequent fly-by of the Sun at a relative speed of more than 300 km/s. So, to get to a speed of 370 km/s instead would require a distance to the Sun of 3.5 instead of 4 solar radii - totally doable if we don't have to deal with sensitive instruments that need to be shielded from the intense radiation and heat.
Now we just have to make sure that the velocity vector of the probe is properly aligned with the CMB, but this is possible: The inclination can be varied by aiming at different edges of the Sun while the direction within the ecliptic just needs a proper timing with respect to the position of Jupiter.
Unfortunately, this maneuver provides the "being at rest w.r.t. the CMB" requirement for the "small experiment" only for one instant in time, and not for an extended period. If you need that, we're back at Jack's answer.
