How to broadcast ARP update to all neighbors in Linux?

Yes, it's called "Unsolicited ARP" or "Gratuitous ARP". Check the manpage for arping for more details, but the syntax looks something like this:

arping -U 192.168.1.101

If you're spoofing an address, you may need to run this first:

echo 1 > /proc/sys/net/ipv4/ip_nonlocal_bind

Finally, because of its spoofing ability, sending Unsolicited ARP packets is sometimes considered a "hostile" activity, and may be ignored, or might lead to being blocked by some third-party firewalls.

What you are looking for is called "Gratuitous ARP" and can be done using "arping". If your IP address is 10.0.0.1 on eth0, you would use this command:

arping -A -i eth0 10.0.0.1

You can verify the ARP is being sent using "tcpdump" while the "arping" is running, in this case I am watching "wlan0":

laptop:~$ sudo tcpdump -lni wlan0 arp    
tcpdump: verbose output suppressed, use -v or -vv for full protocol decode
listening on wlan0, link-type EN10MB (Ethernet), capture size 65535 bytes
12:14:11.219936 ARP, Reply 172.16.42.161 is-at a4:77:03:d2:9b:c4, length 28
12:14:12.220119 ARP, Reply 172.16.42.161 is-at a4:77:03:d2:9b:c4, length 28
12:14:13.220288 ARP, Reply 172.16.42.161 is-at a4:77:03:d2:9b:c4, length 28
12:14:13.220397 ARP, Reply 172.16.42.161 is-at a4:77:03:d2:9b:c4, length 28
^C
3 packets captured
3 packets received by filter
0 packets dropped by kernel
laptop:~$