connect Late 2009 iMac 10.3.6 (21.5" display)with mac mini
Will it be possible to connect my Late 2009 iMac (21.5"" display – High Sierra) with the new Mac Mini M4 chip and keep it as main monitor?
How?
No.
That iMac does not support Target Display Mode. Of the 2009 models, only the 24" and 27" ones did. In addition, Apple placed restrictions on Target Display Mode such that no current Mac will support using any iMac as a Target Display.
A 21.5" iMac would have a resolution of 1920x1080 pixels.
You can find 24" 1920x1080 monitors for anywhere from about $80 (ones with lousy color accuracy) to $150 (ones with IPS panels and near-100% coverage of sRGB). For about $300 – $350, you could find a basic 27" 2560x1440 or 27" 3840x2160 (4K) monitor with an IPS panel and near-100% coverage of sRGB that would be a huge upgrade on that old iMac, would be more convenient to use, and would take less electricity, besides.