How do mRNAs find ribosomes?

The answer to the first part of your question is: diffusion. After transcription and some processing, the eukaryotic mRNA is exported to the cytoplasm. Here it floats around until it meets a ribosome. Binding of ribosome to mRNA is facilitated by proteins called initiation factors (IFs) and the 5' cap of mRNA$^1$.

The second part of your question basically asks about cell-free protein synthesis using an RNA template. This has already been shown to work: one is reminded of the famous Nirenberg–Matthaei experiment$^2$, which ultimately led to the cracking of the genetic code. Nirenberg and Matthaei worked with bacterial cell extracts, but systems based on eukaryotic cell extracts have also been developed$^3$.

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References and further reading:

1. Nelson DL, Cox MM. Lehninger principles of biochemistry. 7th ed (international edition). Basingstoke, England: Macmillan Higher Education; c2017. Chapter 27, Protein metabolism; p. 1077–1126.
2. Nirenberg MW, Matthaei JH. The dependence of cell-free protein synthesis in E. coli upon naturally occurring or synthetic polyribonucleotides. PNAS. 1961 Oct 1;47(10):1588–1602. https://doi.org/10.1073/pnas.47.10.1588
3. Chong S. Overview of cell‐free protein synthesis: historic landmarks, commercial systems, and expanding applications. Curr Protoc Mol Bio. 2014 Oct 1;108(1):16.30.1–11. https://doi.org/10.1002/0471142727.mb1630s108